Handbook of Fractional Calculus with Applications: Volume 7 Applications in Engineering, Life and Social Sciences, Part A 9783110571905, 9783110570915

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Dumitru Baleanu, António Mendes Lopes (Eds.) Handbook of Fractional Calculus with Applications

Handbook of Fractional Calculus with Applications Edited by J. A. Tenreiro Machado

Volume 1: Theory Anatoly Kochubei, Yuri Luchko (Eds.), 2019 ISBN 978-3-11-057081-6, e-ISBN (PDF) 978-3-11-057162-2, e-ISBN (EPUB) 978-3-11-057063-2 Volume 2: Fractional Differential Equations Anatoly Kochubei, Yuri Luchko (Eds.), 2019 ISBN 978-3-11-057082-3, e-ISBN (PDF) 978-3-11-057166-0, e-ISBN (EPUB) 978-3-11-057105-9 Volume 3: Numerical Methods George Em Karniadakis (Ed.), 2019 ISBN 978-3-11-057083-0, e-ISBN (PDF) 978-3-11-057168-4, e-ISBN (EPUB) 978-3-11-057106-6 Volume 4: Applications in Physics, Part A Vasily E. Tarasov (Ed.), 2019 ISBN 978-3-11-057088-5, e-ISBN (PDF) 978-3-11-057170-7, e-ISBN (EPUB) 978-3-11-057100-4 Volume 5: Applications in Physics, Part B Vasily E. Tarasov (Ed.), 2019 ISBN 978-3-11-057089-2, e-ISBN (PDF) 978-3-11-057172-1, e-ISBN (EPUB) 978-3-11-057099-1 Volume 6: Applications in Control Ivo Petráš (Ed.), 2019 ISBN 978-3-11-057090-8, e-ISBN (PDF) 978-3-11-057174-5, e-ISBN (EPUB) 978-3-11-057093-9 Volume 8: Applications in Engineering, Life and Social Sciences, Part B Dumitru Baleanu, António Mendes Lopes (Eds.), 2019 ISBN 978-3-11-057092-2, e-ISBN (PDF) 978-3-11-057192-9, e-ISBN (EPUB) 978-3-11-057107-3

Dumitru Baleanu, António Mendes Lopes (Eds.)

Handbook of Fractional Calculus with Applications |

Volume 7: Applications in Engineering, Life and Social Sciences, Part A Series edited by Jose Antonio Tenreiro Machado

Editors Prof. Dr. Dumitru Baleanu Çankaya University Faculty of Arts and Sciences Department of Mathematics Öğretmenler Caddesi 14 06530 Ankara Turkey [email protected]

Prof. Dr. António Mendes Lopes University of Porto Faculty of Engineering Dept. of Mechanical Engineering Rua Dr. Roberto Frias 4003-465 Porto Portugal [email protected]

Series Editor Prof. Dr. Jose Antonio Tenreiro Machado Department of Electrical Engineering Instituto Superior de Engenharia do Porto Instituto Politécnico do Porto 4200-072 Porto Portugal [email protected]

ISBN 978-3-11-057091-5 e-ISBN (PDF) 978-3-11-057190-5 e-ISBN (EPUB) 978-3-11-057096-0 Library of Congress Control Number: 2019934658 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: djmilic / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Fractional calculus (FC) originated in 1695, nearly at the same time as conventional calculus. However, FC attracted limited attention and remained a purely mathematical exercise in spite of the original contributions of important mathematicians, physicists, and engineers. FC underwent a rapid development during the last few decades, both in mathematics and applied sciences, being nowadays recognized as an excellent tool for describing complex systems and phenomena involving long-range memory effects and non-locality. A huge number of research papers and books devoted to this subject have been published, and each year several specialized conferences and workshops are organized. The popularity of FC in all fields of science is due to its successful application in mathematical models, namely, in the form of FC operators and fractional integral and differential equations. Presently, we are witnessing considerable progress both in theoretical aspects and in applications of FC in areas such as physics, engineering, biology, medicine, economy, and finance. FC has attracted many researchers from all over the world and there is a demand for works covering all areas of science in a systematic and rigorous form. In fact, the amount of literature devoted to FC and its applications is huge, but readers are confronted with a high heterogeneity. The Handbook of fractional calculus with applications (HFCA) intends to fill that gap and provides the readers with a solid and systematic treatment of the main aspects and applications of FC. Motivated by these ideas, the editors of the volumes involved a team of internationally recognized experts for a joint publishing project offering a survey of their own and other important results in their fields of research. As a result of these joint efforts, a modern encyclopedia of FC and its applications, reflecting the core of the present-day scientific knowledge, is now available. This work is distributed as eight distinct volumes, each one developed under the supervision of its editors. The seventh volume of HFCA contains 12 chapters on FC applications in engineering, life sciences, and social sciences. The first group of chapters includes six highlevel contributions in the area of biology and medicine. These chapters discuss the application of fractional differential equations to bio-medical phenomena, the use of empirical models for electro-impedance spectroscopy information, the numerical solutions of singular time-fractional PDEs, the modeling of nociception pathways and pain mechanisms, the use of variable-order derivatives for bone remodeling in the presence of metastases, and the modeling of skeletal muscles by fractional multimodels, including the analysis of length effects. The second group of chapters are contributions in the area of mechanics of complex media. In this group the reader can find six top-level works addressing models for unconfined groundwater, structural mechanics, dynamic problems in viscoelasticity, anomalous solute transport in complex media, the use of variable-order FC models in solid mechanics, and fractional heat conduction models and their applications. https://doi.org/10.1515/9783110571905-201

VI | Preface Our special thanks go to the authors of individual chapters that are excellent surveys of selected classical and new results in several important fields of FC. The editors believe that the HFCA will represent a valuable and reliable reference work for all scholars and professionals willing to develop research in the challenging and timely scientific area. Dumitru Baleanu and António Mendes Lopes

Contents Preface | V Sadia Arshad, Dumitru Baleanu, and Yifa Tang Fractional differential equations with bio-medical applications | 1 António M. Lopes and J. A. Tenreiro Machado Fractional-order modeling of electro-impedance spectroscopy information | 21 Omar Abu Arqub and Shaher Momani Numerical solutions of singular time-fractional PDEs | 43 Clara M. Ionescu, Dana Copot, and Cristina Muresan A multi-scale model of nociception pathways and pain mechanisms | 55 Joana Pinheiro Neto, Duarte Valério, and Susana Vinga Variable-order derivatives and bone remodeling in the presence of metastases | 69 P. Melchior, S. Victor, M. Pellet, J. Petit, J.-M. Cabelguen, and A. Oustaloup Skeletal muscle modeling by fractional multi-models: analysis of length effect | 95 Hossein Jafari, Behrouz Mehdinejadiani, and Dumitru Baleanu Fractional calculus for modeling unconfined groundwater | 119 Yury Rossikhin and Marina Shitikova Fractional calculus models in dynamic problems of viscoelasticity | 139 Yury Rossikhin and Marina Shitikova Fractional calculus in structural mechanics | 159 HongGuang Sun, Yong Zhang, and Wen Chen Anomalous solute transport in complex media | 193 Behrouz Parsa Moghaddam, Arman Dabiri, and José António Tenreiro Machado Application of variable-order fractional calculus in solid mechanics | 207

VIII | Contents Jan Terpak Fractional heat conduction models and their applications | 225 Index | 247

Sadia Arshad, Dumitru Baleanu, and Yifa Tang

Fractional differential equations with bio-medical applications

Abstract: In this chapter, we investigate the dynamics of fractional order models in bio-medical. First, we examine the fractional order model of HIV Infection and analyze the stability results for non-infected and infected equilibrium points. Then, we concentrate on the fractional order tumor growth model and establish a sufficient condition for existence and uniqueness of the solution of the fractional order tumor growth model. Local stability of the four equilibrium points of the model, namely the tumor free equilibrium, the dead equilibrium of type 1, the dead equilibrium of type 2 and the coexisting equilibrium is investigated by applying Matignons condition. Dynamics of the fractional order tumor model is numerically investigated by varying the fractional-order parameter and the system parameters. Keywords: Fractional calculus, bio-medical models, stability analysis, numerical simulations MSC 2010: 34A08, 34A34, 92B05, 65L12, 65L20

1 Introduction The study of fractional calculus started at the end of the seventeenth century. It is a branch of mathematical analysis in which integer-order derivatives and integrals extend to a real or complex number. In the end of the nineteenth century basic theory of fractional calculus was developed with the studies of Liouville, Grünwald, Letnikov, and Riemann. We refer to [9] and [17] for the basic theory of fractional differential equations. Recently, fractional differential equations have been widely used in various fields such as muscular blood vessel modeling [1], non-linear oscillation of earthquakes [11], fluid-dynamic traffic modeling [15], biotechnology [19], continuum and statistical mechanics [20], and global economy models [29]. Fractional derivatives are useful in rheology as they represent crucial features of cell rheological behavior [10]. Also, describing the behavior of brainstem vestibular-oculomotor neurons by fractional ordinary differential equations is more reasonable than by classical Sadia Arshad, Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan, e-mail: [email protected] Dumitru Baleanu, Department of Mathematics, Cankaya University, Ankara 06530, Turkey; and Institute of Space Sciences, Magurele-Bucharest, Romania, e-mail: [email protected] Yifa Tang, LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, e-mail: [email protected] https://doi.org/10.1515/9783110571905-001

2 | S. Arshad et al. integer-order models [3]. In biology, it has been deduced that the membranes of cells of biological organisms have fractional-order electrical conductance [5]; hence some mathematical models which describe cell behavior are classified into groups of noninteger-order models. The purpose of biological models is to explore the vital dynamics of the system and find the answers of particular questions about that model. We examine the fractional-order model of three types of populations of cells, that is, productively infected T cells, I, non-infected activated CD4+ T cells, T, and HIV virus particles, V. Tumor growth is a process involving various types of cells that interact in a complex way. A wide range of mathematical biological models exists presently, each of which focus on one or two features of the complex process of the tumor growth and the treatment strategy. However, present models are not able to describe various clinically identified findings. First we will investigate a cancer growth model which includes numerous key components of the tumor growth and the consequence of their interactions. We apply a discretization process to the fractional-order tumor model introduced in [12].

2 Dynamical investigations of fractional-order HIV model In this section, we study the cycling CD4+ T cell model of HIV infection of arbitrary order. In HIV-infected patients with CD4+ T cell count, the distribution rate of CD4+ T cells declines more or less linearly, suggesting that the expansion rate is densitydependent and is managed by a logistic-like growth function. Clinical data from diverse sources suggest that the availability of target cells restricts virus replication. Therefore we investigate the dynamics of our model by varying the viral load in blood by finding equilibrium points of our model. We establish the stability results for noninfected and infected equilibrium points. To illustrate the effect of this dynamics, we present the numerical simulations of the fractional model. The HIV virus provides a basis to develop acquired immune deficiency syndrome (AIDS). For HIV-1 infection the primary target is an activated CD4+ T cell [13]. The availability of target cells is a significant control factor, i. e., cells that HIV is capable to infect [4, 6]. CD4+ T cells are infected by the virus upon HIV infection. Clinical data from diverse sources suggest that the availability of CD4+ T cells restricts virus replication. An infected CD4+ T cell makes multiple HIV copies and does not affect its function in the human immune system. A shortage of CD4+ T cells affects the function of the human immune system, inhibiting it to work appropriately. The viral load is raised by activating the immune system with IL-2 [18]. A durable control of HIV-1 can be accomplished by monotherapy using the anti-retroviral drug didanosine or by combined

Fractional differential equations with bio-medical applications | 3

therapy with the immunosuppressive drug hydroxyurea [30]. Literature on target cell– limited models predicted this long-term effect previously [6]. The aim of this section is to study the dynamics of the HIV model for fractional-order derivatives. We examine the stability of fractional-order models of three types of cell populations, i. e., productively infected T cells, I, non-infected activated CD4+ T cells, T, and HIV virus particles, V. First, we outline some definitions used throughout the remaining sections. The Riemann–Liouville fractional integral I q u of order q > 0 of u : ℝ+ → ℝ is defined by t

1 I u(t) = ∫(t − s)q−1 u(s)ds, Γ(q) q

0

provided the expression on the right-hand side is defined. Here Γ denotes the Gamma function. The Caputo fractional derivative Dq u of order q of a continuous function u : ℝ+ → ℝ is defined by 1

t

∫0 (t − s) { Dq u(t) = { Γ(m−q) m d u(t) { dt m ,

m−q−1 (m)

u

(s)ds,

m − 1 ≤ q < m, m = q.

In particular, when 0 < q < 1, we have Dq u(t) =

t

1 ∫(t − s)−q u󸀠 (s)ds. Γ(1 − q) 0

Our aim is to examine the fractional-order 0 < q ≤ 1 model that includes numerous types of cells: non-infected activated CD4+ T cells, T, infected T cells, I, and HIV-1 virus particles, V, proposed by Bore and Perelson [7]. The interactions between these cells can be defined by the following differential equations: Dq T = αTq T(1 − Ttot /Tmax ) − (βq + γ q )TV, { { q D I = βq TV − δIq I, { { q q q {D V = p I − c V,

t ≥ 0, t ≥ 0, t ≥ 0,

(1)

with the initial conditions T(0) = T0 , I(0) = I0 , V(0) = V0 , where T0 , I0 , V0 ∈ ℝ+ . In HIV-infected patients with CD4+ T cell count, the expansion rate is densitydependent because the distribution rate of CD4+ T cells declines more or less linearly and is managed by a logistic-like growth function. The CD4+ T cell count is around 1000 CD4+ T cells per μL in an uninfected individual. It is difficult to estimate the maximum rate of growth, but an upper approximation is given during treatment by the rate of recovery, i. e., about 0.1 per day [16]. For infected T cells the average lifetime is between 1 and 2 days approximately [16, 27, 26] and for virus particles the average lifetime is estimated to be 8 hr maximally [27].

4 | S. Arshad et al. Table 1: HIV model parameters. Parameters

Description

Values

β γ p αT Tmax δI c

Infection rate Depletion of CD4+ T cells Production rate of infected cells Maximum pace of T renewal Non-infected steady state Turnover pace of prolifically infected T cells Clearance pace of virus particles

7.6 × 10

virions day−1 7.2 × 10−4 day−1 100 day−1 0.1 day−1 1000 cells μL−1 0.5 day−1 3 day−1

−5

−1

Here β represents the infection rate and γ represents the virus-induced decrease of all types of the CD4+ T cells. The causes could range from apoptosis [23] to destruction of lymphoid tissue [25, 24]. The highest rate of T cell restoration is represented by the increasing rate αT = 0.1 day−1 , the non-infected steady-state CD4+ count is Tmax = 1000 cells μL−1 , and the turnover rate is δI = 0.5 day−1 for infected T cells. Note that Ttot = T + I represents the entire number of T cells (the CD4+ count). Computing CD4+ T cell counts in the patient blood as well as viral loads is generally used to assess the clinical status of a patient. The virus and cell counts in a typical patient are around 105 virions per μL of plasma and 200 CD4+ cells per μL. In our model, infected cells, I, produce virus particles, V, at pace p per cell and are cleaned at a pace c (per capita). Adhering of elements to cells and support by antibodies may increase the clearance rate constant c. The dynamics of cells [27] (set at c ≥ 3 day−1 ) are not quicker than the dynamics of the viral molecules. Per day the production rate per cell p has been estimated to be approximately p = 100 [14] virions. The parameters with their values are listed in Table 1.

2.1 Equilibrium points and stability To find the equilibrium points of (1), we solve the following system: Dq T = 0, { { q D I = 0, { { q D { V = 0.

(2)

After solving system (2) we obtain two equilibrium points, the contamination-free equilibrium E0 = (T0 , I0 , V0 ) = (Tmax , 0, 0) and the infected equilibrium E = (T, I, V), where cq δIq T = q q, p β I=

cq αTq (pq βq Tmax − cq δIq )

pq βq (αTq cq + pq (βq + γ q )Tmax )

,

Fractional differential equations with bio-medical applications | 5

V=

pq I. cq

Now we give a detailed stability analysis of the non-infected steady state E0 and the infected steady state E. The Jacobian matrix at equilibrium point (T ∗ , I ∗ , V ∗ ) is given by q q ∗ 󵄨󵄨 αq − αq (2T ∗ + I ∗ )/T 󵄨󵄨 T max − (β + γ )V T 󵄨󵄨 q ∗ J(M) = 󵄨󵄨󵄨 β V 󵄨󵄨 󵄨󵄨 0

−αTq T ∗ /Tmax −δIq pq

−(βq + γ q )T ∗ 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 . βq T ∗ 󵄨󵄨 󵄨󵄨 −cq 󵄨

(3)

The associated characteristic polynomial equation is given by |J(M) − λE| = 0, where E is the identity matrix. To analyze the local asymptotic stability of the noninfected steady state E0 , we get the following result. Theorem 2.1. Let βq < totically stable.

δIq cq . pq Tmax

Then the non-infected steady state E0 is locally asymp-

Proof. The characteristic polynomial at E0 = (T0 , 0, 0) is given by (−αTq − λ).((−δIq − λ)(−cq − λ) − pq βq T0 ) = 0.

(4)

Clearly λ1 = −αTq < 0. Next we consider the polynomial λ2 + (δIq + cq )λ − δIq cq − pq βq Tmax = 0.

(5)

Solving equation (5), we get 1 2 λ2 = (−(δIq + cq ) − √(δIq − cq ) + 4pq βq Tmax ) 2 and 1 2 λ3 = (−(δIq + cq ) + √(δIq − cq ) + 4pq βq Tmax ). 2 As all the parameters are positive, λ2 < 0 and λ3 < 0 if βq
0, A1 > 0, A3 > 0, and A1 A2 > A3 , (ii) P(λ) < 0, A1 ≥ 0, A2 ≥ 0, A3 > 0, and q < 32 , (iii) P(λ) < 0, A1 < 0, A2 < 0, and q > 32 .

2.2 The Grünwald–Letnikov approximation Suppose that u : [0, T] → R is a smooth function (0, t) with t ≤ T and 0 < q < 1. Choose a uniform grid on [0, t] 0 = t0 < t1 < ⋅ ⋅ ⋅ < tn+1 = t = (n + 1)τ with tn+1 − tn = τ. The Grünwald–Letnikov (GL) derivative is defined by Dq u(t) = lim

τ→0

1 q Δ u(t), τq τ

where n+1 1 q 1 Δ u(t) = (u(t ) − ∑ wkq u(τn+1−k )) n+1 τq τ τq k=1

and

q wkq = (−1)k−1 ( ) . k These binomial coefficients are given [28] by wkq = (1 −

where the first coefficient is w1q = q.

q+1 q )wk−1 , k

Fractional differential equations with bio-medical applications | 7

Suppose that following fractional differential equation has a unique solution u = u(s) in [0, T]: Dq u(t) = f (u(t)), u(t0 ) = u0 . Let uk ≈ u(tk ). Choose a uniform grid 0 = t0 < t1 < ⋅ ⋅ ⋅ < tN+1 = T,

with tk+1 − tk = τ.

Then, by using [28], we arrive at n+1

q u0 , un+1 = τq f (un ) + ∑ wkq u(tn+1−k ) + rn+1 k=1

q is the correction term defined by where rn+1

rkq =

t k+1 . Γ(k + 1 − q)

Let Tk , Ik , and Vk denote the numerical approximation of T(tk ), I(tk ), and V(tk ), respectively. Then we get the following numerical approximation based on the GL method for fractional model (1): n+1

q { { Tn+1 = τq (αTq Tn (1 − (Tn + In )/Tmax ) − (βq + γ q )Tn Vn ) + ∑ wkq Tn+1−k + rn+1 T0 , { { { { k=1 { { n+1 { { q q q q I = τ (β T V − δ I0 , I ) + ∑ wkq In+1−k + rn+1 n+1 n n n I { { { k=1 { { { n+1 { { q { {Vn+1 = τq (pq In − cq Vn ) + ∑ wkq Vn+1−k + rn+1 V0 . { k=1

2.3 Numerical simulations In this section, we demonstrate results on the stability of equilibrium points by using numerical solutions. We choose the infection rate βq = 0.000012q of infected cells δq c q

and, using the rest of the parameter values given in Table 1, we obtain βq < pq TI . max Thus by Theorem 2.1, the non-infected equilibrium E0 is locally asymptotically stable. Numerical solutions are in good agreement with our analytical result (Theorem 2.1); see Figure1 for q = 0.7 and q = 1. If we increase βq = 0.00005q and use the remaining values from Table 1, all the conditions of Theorem 2.2 are satisfied. Therefore the infected equilibrium E is locally asymptotically stable. Numerical solutions demonstrate that the trajectories of system (1) come close to the equilibrium E (Figure 2).

8 | S. Arshad et al.

Figure 1: Simulation results of non-infected activated CD4+ T cells, T , infected T cells, I, and virus particles, V , with initial condition (800, 10, 10).

Now we fix βq = 0.00015q , pq = 105q , and Tmax = 100 and use the remaining values from Table 1. Then, the infected equilibrium E = is locally asymptotically stable by Theorem 2.2. Numerical solutions of system (1) move toward the steady state E (Figure 3). Next we set βq = 0.00015q and Tmax = 900 and use the remaining values from Table 1. Then the infected equilibrium E = is locally asymptotically stable for q < 2/3 by Theorem 2.2, condition (ii). Numerical simulations demonstrate that the trajectories of system (1) move toward the steady state E for q = 0.6, whereas the system is not stable for q = 1 (Figure 4).

2.4 Discussion In summary, we have investigated the stability conditions of a model of HIV infection of CD4+ T cells, infected T cells, and HIV virus particles with differential equations. We have established the stability properties of the equilibrium point. Based on the

Fractional differential equations with bio-medical applications | 9

Figure 2: Simulation results of non-infected activated CD4+ cells, T , infected T cells, I, and virus particles, V , with initial condition (800, 1, 1) for q = 1 and q = 0.7.

parameters, we obtain a sufficient condition for the stability of the uninfected equilibrium point. Moreover, we have analytically derived the stability criteria for the infected equilibrium point by analyzing the transcendental characteristic equation. We have studied the dynamics of the model by varying the infection rate β for different values of the fractional order. It is clear from the presented figures that our analytical results are in good agreement with results obtained by using numerical simulations.

3 Dynamical analysis of fractional-order cancerous tumor model In this section, we examine a fractional-order cancer model that incorporates the interactions between tumor cells, immune cells, and normal cells. We prove a sufficient condition for existence and uniqueness of the solution of the fractional-order cancer model. By using Matignons conditions, we study the local stability of the four equi-

10 | S. Arshad et al.

Figure 3: Simulation results of non-infected activated CD4+ cells, T , infected T cells, I, and virus particles, V , with initial condition (200, 5, 10).

librium points of the model. Further, numerical simulations are presented that are in good agreement with our theoretical results by varying the fractional order q ∈ (0, 1]. Matignon’s results [21] can be used to find the local stability of the equilibrium points of a linearized fractional-order system. We have |arg(λi )| > qπ/2 (i = 1, 2, 3),

(7)

where λ1 , λ2 , λ3 are the eigenvalues of the Jacobian matrix evaluated at the equilibrium points. Consider the following fractional-order system: Dq U(t) = f (U(t)),

U(0) = U0 ,

(8)

where U(t) = (u1 , u2 , u3 )T ∈ R3 and f : R3 → R3 is a vector function in terms of U. The Jacobian matrix computed at U ∗ = (u∗1 , u∗2 , u∗3 ) is J(U ∗ ) = (

𝜕fi 󵄨󵄨󵄨󵄨 ) 󵄨 . 𝜕uj ij 󵄨󵄨󵄨U=U ∗

The following lemma will be useful to establish stability results [22].

(9)

Fractional differential equations with bio-medical applications | 11

Figure 4: Simulation results of non-infected activated CD4+ T cells, T , infected T cells, I, and virus particles, V , with initial condition (120, 10, 90).

Lemma 3.1. If all the eigenvalues λ1 , λ2 , λ3 of the equilibrium points U ∗ of system (8) satisfy Matignon’s conditions (7), then U ∗ is locally asymptotically stable. In [8] a mathematical model has been proposed of interactions among tumor cells, immune cells, and normal cells, given by the following differential equations: ρℐ𝒯 dℐ { =s+ − c1 ℐ𝒯 − dℐ , { { dt α +𝒯 { { { { d𝒯 = r1 𝒯 (1 − b1 𝒯 ) − c2 ℐ𝒯 − c3 𝒯 𝒩 , { { dt { { { { { d𝒩 = r2 𝒩 (1 − b2 𝒩 ) − c4 𝒯 𝒩 , { dt

t ≥ 0, t ≥ 0,

(10)

t ≥ 0,

where ℐ (t) denotes the number of immune cells, 𝒯 (t) the number of tumor cells, and 𝒩 (t) the number of normal cells at time t. The parameter values are summarized in Table 2. In this section we are concerned with the generalization of the integer-order model given in (10) to the fractional order q ∈ (0, 1] of interactions among tumor cells, im-

12 | S. Arshad et al. Table 2: Parameter values for cancer model. Parameters

Description

Values

s ρ α b1 b2 r1 r2 c1 c2 c3 c4 d

Immune source rate Immune response rate Immune threshold rate Carrying capacity of tumor cells Carrying capacity of normal cells Per capita growth rate of tumor cells Per capita growth rate of normal cells Competition term Competition term Competition term Competition term Per capita death rate of immune cells

0 ≤ s ≤ 0.5 0 ≤ ρ ≤ 2.5 0.3 10 1 2 1 0.4 2 0.5 1.5 0.2

mune cells, and normal cells. We have ρq ℐ𝒯 q q { D ℐ (t) = s + − c1q ℐ𝒯 − dq ℐ , { { { αq + 𝒯 { { Dq 𝒯 (t) = r1q 𝒯 (1 − bq1 𝒯 ) − c2q ℐ𝒯 − c3q 𝒯 𝒩 , { { { { q q q q {D 𝒩 (t) = r2 𝒩 (1 − b2 𝒩 ) − c4 𝒯 𝒩 ,

t ≥ 0,

(11)

t ≥ 0, t ≥ 0,

together with the initial conditions ℐ (0) = ℐ0 , 𝒯 (0) = 𝒯0 , 𝒩 (0) = 𝒩0 .

3.1 Existence and uniqueness and stability analysis of fractional-order cancer model The fractional-order tumor model (11) can be written in the following form: Dq U(t) = F(U(t)),

t ∈ (0, T],

U(0) = U0 ,

(12)

where ℐ [ ] U = [𝒯 ] , [𝒩 ]

ℐ0 [ ] U0 = [ 𝒯0 ] , [𝒩0 ]

F(U) =

ρq ℐ𝒯 − c1q ℐ𝒯 − dq ℐ αq +𝒯 [ [r q 𝒯 (1 − bq 𝒯 ) − cq ℐ𝒯 − cq 𝒯 [1 1 3 2 r2q 𝒩 (1 − bq2 𝒩 ) − c4q 𝒯 𝒩 [

sq +

Define the supremum norm as ‖F‖ = sup |F(t)|. t∈(0,T]

] ]

𝒩].

]

Fractional differential equations with bio-medical applications | 13

The norm of the matrix M = [mij (t)] is defined by ‖M‖ = Σi,j sup |mij (t)|. t∈(0,T]

Theorem 3.2. System (11) with the initial condition U(0) = U0 has a unique solution in the region Δ × (0, T] if 1

2ρq μq 1− μqq Tq e α ), max{(dq + c1q μq + Γ(q + 1) αq (r1q + 2r1q bq1 μq + c2q μq + c3q μq ), (r2q + 2r2q bq2 μq + c4q μq )} < 1,

(13)

where t ∈ (0, T], Δ = {(ℐ , 𝒯 , 𝒩 ) : max(|ℐ |, |𝒯 |, |𝒩 |)} ≤ μ. Proof. The solution of system (11) has the following form: t

1 V = U0 + ∫(t − s)q−1 F(U(s))ds = Φ(U). Γ(q) 0

This gives t

Φ(U1 ) − Φ(U2 ) =

1 ∫(t − s)q−1 (F(U1 (s)) − F(U2 (s)))ds. Γ(q) 0

Hence, we have |Φ(U1 ) − Φ(U2 )| 󵄨t 󵄨󵄨 󵄨󵄨 1 󵄨󵄨󵄨󵄨 q−1 = 󵄨󵄨∫(t − s) (F(U1 (s)) − F(U2 (s)))ds󵄨󵄨󵄨 󵄨󵄨 Γ(q) 󵄨󵄨󵄨 󵄨 0 t

1 ≤ ∫(t − s)q−1 |(F(U1 (s)) − F(U2 (s)))|ds Γ(q) 0

1

2ρq μq 1− μqq T max{(dq + c1q μq + e α ), ≤ Γ(q + 1) αq q

(r1q + 2r1q bq1 μq + c2q μq + c3q μq ), (r2q + 2r2q bq2 μq + c4q μq )}‖U1 − U2 ‖ = L‖U1 − U2 ‖, where 1

L=

2ρq μq 1− μqq Tq max{(dq + c1q μq + e α ), Γ(q + 1) αq (r1q + 2r1q bq1 μq + c2q μq + c3q μq ), (r2q + 2r2q bq2 μq + c4q μq )}.

14 | S. Arshad et al. Thus, if L < 1, then the mapping Φ(U) is a contraction mapping and this yields that system (11) has a unique solution in the region Δ × (0, T].

3.2 Equilibrium points To find equilibrium points of system (11), we solved the system described by the following equations: Dq ℐ (t) = 0,

Dq 𝒯 (t) = 0,

Dq 𝒩 (t) = 0.

There are the following three types of equilibrium points: – Tumor-free equilibrium: E0 = (sq /dq , 0, 1); in this category, the tumor cells are zero but the normal cells survive. – Dead equilibrium: 1. type 1: E1 = (sq /dq , 0, 0); in this category, both tumor cells and normal cells died off; 2. type 2: E2 = (f (x), x, 0); here the normal cells alone have died off and the tumor cells have survived, where x is a non-negative solution to x + (c2q /r1q bq1 )f (x) − 1/bq1 = 0 and f (x) = –

sq (αq + x) . c1q x(αq + x) + dq (αq + x) − ρq x

(14)

(15)

Co-existing equilibrium: E3 = (f (y), y, g(y)); here both normal and tumor cells coexist, where y is a non-negative solution of y + (c2q /r1q bq1 )f (y) + (c3q /r1q bq1 )g(y) − 1/bq1 = 0 and g(y) = 1 − (

c4q r2q

)y.

(16)

(17)

There may exist zero to three dead equilibria of type 2 based on the parameter values. Equation (14) gives a third-order polynomial for the x values of the dead equilibriums of type 2, i. e., B1 x3 + B2 x2 + B3 x + B4 = 0, where B1 = c1q , B2 = c1q αq + dq − ρq −

c1q

bq1

,

Fractional differential equations with bio-medical applications | 15

c1q αq

B3 = dq αq −

bq1

c2q sq αq

B4 =

−

r1q bq1

−

q q dq ρq c2 s q + q + q q, b1 b1 b1 r1

d q αq . r1q bq1

Similarly there exist zero to three co-existing equilibrium points, depending on the parameter values. The following third-order polynomial for the y values has obtained from equation (16) for the co-existing equilibrium: C1 y3 + C2 y2 + C3 y + C4 = 0, where C1 = c1q r2q − C2 = C3 = C4 =

c1q c3q c4q r1q bq1

,

c3q c4q q q q q q q + (r2 − q q )(c1 α + d − ρ ), b1 r1 b1 q q cq r q r q c q r q sq c c d q α q dq r2q αq + 2q 2 q − 3 4q q + ( q3 2q − 2q )(c1q αq r1 b1 r1 b1 r1 b1 b1 q q q q q q q q q q q c r d α r d α c2 r2 s α + 3 2q q − 2 q . q q r1 b1 r1 b1 b1 c1q c3q r2q r1q bq1

−

c1q r2q

+ dq − ρq ),

The Jacobian matrix of system (11), evaluated at equilibrium point (ℐ ∗ , 𝒯 ∗ , 𝒩 ∗ ), is given by J(ℐ ∗ , 𝒯 ∗ , 𝒩 ∗ ) [ =[ [

ρq 𝒯 ∗ αq +𝒯 ∗

− c1q 𝒯 ∗ − dq

−c2q 𝒯 ∗ 0

r1q −

α q ρq ℐ ∗ − c1q ℐ ∗ (αq +𝒯 ∗ )2 2r1q bq1 𝒯 ∗ − c2q ℐ ∗ − c3q 𝒩 ∗ −c4q 𝒩 ∗

0 ] ]. −c3q 𝒯 ∗ q q ∗ q ∗ q r2 − 2r2 b2 𝒩 − c4 𝒯 ] (18)

3.3 Stability analysis of equilibrium points Theorem 3.3. The tumor-free equilibrium point E0 = (sq /dq , 0, 1) of system (11) is locally

asymptotically stable if r1q
0, G1 > 0, G3 > 0, and G1 G2 > G3 , (ii) P(λ) < 0, G1 ≥ 0, G2 ≥ 0, G3 > 0, and q < 32 , (iii) P(λ) < 0, G1 < 0, G2 < 0, and q > 32 .

3.4 Numerical simulations To illustrate the theoretical results obtained in Section 3.3, we present numerical simulations. For the parameter values given in Table 2, we choose sq = 0.3q , ρq = 0.6q , and the tumor-free equilibrium E0 is asymptotically stable. Numerical simulations are displayed in Figure 5 with initial condition (ℐ0 , 𝒯0 , 𝒩0 ) = (1, 0.001, 2).

Figure 5: Simulated trajectories for stable tumor-free equilibrium.

18 | S. Arshad et al.

Figure 6: Simulated trajectories for stable tumor-free equilibrium.

Figure 7: Unstable co-existing equilibrium for model (11) for q = 0.4, 0.5, 0.7, 1.

Now we take sq = 0.15q , ρq = 0.6q , c1q = 5q , dq = 0.2q , r1q = 6q , and the rest of the values are as given in Table 2. Then the co-existing equilibrium E4 is asymptotically stable by Theorem 3.5. Figure 6 shows numerical simulations with initial condition (ℐ0 , 𝒯0 , 𝒩0 ) = (0.5, 0.01, 1).

Fractional differential equations with bio-medical applications | 19

Next we fix the parameters sq = 0.001q , ρq = 1q , bq1 = 1.5q , and the remaining values are as given in Table 2. Then it is not asymptotically stable by Theorem 3.5. In Figure7 numerical simulations are displayed with initial condition (ℐ0 , 𝒯0 , 𝒩0 ) = (0.01, 0.05, 1).

3.5 Discussion In conclusion, the dynamics of the fractional-order q ∈ (0, 1) model of interactions among tumor cells, normal cells, and immune cells has been studied. We have studied the existence and uniqueness of the solution of the tumor growth model. Dynamics of the fractional system is studied and stability results of equilibrium points are investigated. Further dynamics of the fractional-order tumor model is numerically examined by varying the system parameters sq , ρq , c1q , bq1 , r1q and the fractional order parameter q. The numerical results are in good agreement with the analytical predictions, indicating the validity of the numerical and theoretical analyses.

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[14] A. T. Haase, K. Henry, M. Zupancic, G. Sedgewick, R. A. Faust, and H. Melroe, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274 (1996), 985–989. [15] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. Soc., 15 (1999), 86–90. [16] D. D. Ho, A. Neumann, A. S. Perelson, W. Chen, J. M. Leonard, and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123–126. [17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, New York, NY, USA, 2006. [18] J. A. Kovacs, M. Baseler, R. J. Dewar, S. Wogel, R. T. J. Davey, and J. Falloon, Increases in CD4 T lymphocytes with intermittent courses of interleukin-1 in patients with human immunodeficiency virus infection: A preliminary study, N. Engl. J. Med., 332 (1995), 567–575. [19] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1–377. [20] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, 291–348, 1997. [21] D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational Engineering in Systems and Application Multi-Conference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, pp. 963–968, July 1996. [22] A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol–Duffing circuit, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 975–986. [23] L. Meyaard, S. A. Otto, R. R. Jonker, M. J. Mijnster, R. P. Keet, and F. Miedema, Programmed death of T cells in HIV-1 infection, Science, 257 (1992), 217–219. [24] G. Pantaleo, How immune-based interventions can change HIV therapy, Nat. Med., 3 (1997), 483–486. [25] G. Pantaleo, C. Graziosi, J. F. Demarest, L. Butini, M. Montroni, and C. H. Fox, HIV infection is active and progressive in lymphoid tissue during the clinically latent stage of disease, Nature, 362 (1993), 355–358. [26] A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, and K. Saksela, Decay characteristics of HIV-1 infected compartments during combination therapy, Nature, 387 (1997), 188–191. [27] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D. D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582–1586. [28] R. Scherer, S. L. Kalla, Y. Tang, and J. Huang, The Grünwald–Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902–917. [29] J. A. Tenreiro Machado and M. E. Mata, Pseudo phase plane and fractional calculus modeling of western global economic downturn, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 396–406. [30] J. Vila, F. Biron, F. Nugier, T. Vallet, and D. Peyramond, 1 year follow-up of the use of hydroxycarbamide and didanosine in HIV infection, Lancet, 348 (1996), 203–204.

António M. Lopes and J. A. Tenreiro Machado

Fractional-order modeling of electro-impedance spectroscopy information Abstract: This chapter applies the electrical impedance spectroscopy technique to characterize objects, such as plant leaves, vegetables, wine, and milk. In a first phase, the impedance is measured and the tools of fractional calculus are applied to model the samples. In a second phase, the model parameters are correlated with known information about the materials. It is verified that fractional-order modeling tools are able to conveniently identify the impedance data with a limited number of parameters. Keywords: electrical impedance spectroscopy, fractional calculus, modeling, visualization PACS: 89.20.-a, 89.75.-k, 07.07.-a

1 Introduction Electrical impedance spectroscopy (EIS) measures the electrical impedance of a specimen across a given range of frequencies [21, 27, 28, 4, 19, 30]. This technique has the advantage of being non-destructive, while avoiding complex and time consuming experimental or laboratory procedures. EIS has been widely used for studying vegetable tissues [49, 41, 3, 8, 24, 47], animal and human samples [52, 12, 1, 16, 23], beverages [53, 42], non-biological materials [50, 20, 14], and devices [15, 22, 2]. In practical terms, EIS involves exciting a specimen with frequency-variable electric sinusoidal signals and registering the system response. The voltage v(t) and current i(t) across the specimen at steady state are sinusoidal functions of time given by v(t) = V cos(ωt + θV ), { i(t) = I cos(ωt + θI ),

(1)

where {V, I} are the amplitudes of the voltage and current, {θV , θI } denote their phase shifts, ω = 2πf represents the angular frequency, and f is the frequency. António M. Lopes, UISPA–LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, e-mail: [email protected] J. A. Tenreiro Machado, Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal, e-mail: [email protected] https://doi.org/10.1515/9783110571905-002

22 | A. M. Lopes and J. A. T. Machado The voltage and current can be represented in the frequency domain by: V(jω) = V ⋅ ejθV , { I(jω) = I ⋅ ejθI ,

(2)

where j = √−1. The complex impedance Z(jω) is defined as the ratio of phasors, i. e., Z(jω) =

V(jω) V j(θV −θI ) = ⋅e = |Z(jω)| ⋅ ej arg [Z(jω)] . I(jω) I

(3)

This chapter addresses the application of EIS to different materials, namely, plant leaves, vegetables, wine, and milk [27–29, 31]. In a first phase, the impedance Z(jω) is measured and the tools of fractional calculus (FC) [34, 33] are applied to model the samples. In a second phase, the model parameters are correlated with known information about the materials. It is verified that fractional-order (FO) models offer a tool to conveniently identify the impedance data with a limited number of parameters. This chapter is organized as follows. Section 2 introduces some fundamental mathematical concepts. Section 3 describes the impedance spectra, Z(jω), by means of FC tools and correlates the model parameters with other known data. Finally, Section 4 draws the main conclusions.

2 Empirical fractional-order models Given an impedance spectrum obtained from experimental data, it is necessary to find a mathematical description, that is, a “model” that fits well into the data and has a limited number of parameters [37, 26, 38]. Different empirical models developed in the context of the dielectric relaxation phenomenon have been used for that purpose [11]. Electromagnetic phenomena are described by Maxwell’s equations, which relate electric and magnetic variables. As all materials used in this study have weak diamagnetic properties, these are neglected and just the effect of an electric field applied to the medium is considered [48]. Thus, the dielectric displacement, D, is related to the electric field, E, and to the polarization density, P, according to D = ε0 E + P,

(4)

where ε0 is the vacuum permittivity. The polarization density of the material, P, is defined as the induced dipole moment per unit volume. If the dielectric is isotropic and uniform, the polarization is assumed to be proportional to the electric field, i. e., P = ε0 χE, with χ representing the dimensionless dielectric susceptibility.

(5)

Fractional-order modeling of electro-impedance spectroscopy information |

23

The combination of (4) and (5) yields D = ε0 (1 + χ)E = ε0 εE,

(6)

where ε = 1 + χ is the relative, dimensionless, dielectric constant. In general, a material cannot be instantaneously polarized in response to an applied electric field and different time scales are associated to different polarization mechanisms [9]. During dielectric polarization, forces are applied on the electric charges in the material, causing charge displacements from their average equilibrium positions. Positive and negative charges slightly move in opposite directions, originating an internal electric field that reduces the overall field within the material. After a certain period of time an equilibrium condition is reached. Analogously, when the external electric field vanishes, the electric charges tend to return to their original locations due to thermally induced motions. Thus, the dielectric polarization decays over time, according to the same function of the material polarization. This phenomenon is known as dielectric relaxation. In analytical terms, the dielectric displacement, D(t), caused by a time variant electric field, E(t), is given by [9] t

D(t) = ε∞ E(t) + ∫ −∞

dΦ(u) ⋅ E(t − u)du, du

(7)

where Φ(t) = (εs − ε∞ )[1 − φ(t)] is the dielectric response function. The quantities (εs , ε∞ ) denote the low- and high-frequency, f = ω/2π, limits of the complex dielectric permittivity, ε∗ (jω) = ε+σ/jωε0 . The parameter σ represents the electrical conductivity of the material. If the complex electrical conductivity, σ ∗ , is introduced, then σ ∗ = σ + jωε0 ε, which takes into account dielectric losses or imperfect conductor materials. The complex dielectric permittivity ε∗ (jω) is related to the relaxation function, φ(t), by ε∗ (jω) − ε∞ dφ(t) ], = ℱ [− εs − ε∞ dt

(8)

where, ℱ [⋅] denotes the Fourier transform. When the relaxation function is an exponential law, i. e., φ(t) = e−t/τ ,

(9)

the well-known Debye equation is obtained, where τ represents the relaxation time. Then we have ε∗ (jω) = ε∞ +

εs − ε∞ . 1 + jωτ

(10)

24 | A. M. Lopes and J. A. T. Machado The Debye model does not adequately describe many experimental data originated from complex materials. On the contrary, such data are better described by nonexponential relaxation laws. This anomalous dielectric behavior can be represented in terms of a continuous distribution of relaxation times, G(τ), meaning that ∞

ε (jω) = ε∞ + (εs − ε∞ ) ⋅ ∫ ∗

0

G(τ) dτ, 1 + jωτ

(11)

where ∫0 G(τ)dτ = 1. The corresponding expression for the relaxation function is ∞

∞

t

φ(t) = ∫ G(τ) ⋅ e(− τ ) dτ.

(12)

0

Materials that deviate from the classical exponential Debye pattern (10) can often be described by the Havriliak–Negami relationship, i. e., ε∗ (jω) = ε∞ +

εs − ε∞

[1 + (jωτ)α ]β

,

0 < α ≤ 1, 0 < β ≤ 1,

(13)

which yields [43] ε̃HN (jω) =

ε∗ (jω) − ε∞ 1 , = εs − ε∞ [1 + (jωτ)α ]β

0 < α ≤ 1, 0 < β ≤ 1,

(14)

where ε̃ denotes the complex susceptibility. Herein α and β are empirical exponents. For α = β = 1 the Debye model is obtained, ε̃D (jω), for α ≠ 1 and β = 1 the Cole–Cole equation, ε̃CC (jω), results, and for α = 1 and β ≠ 1 the Cole–Davidson model, ε̃CD (jω), holds, so that we have ε̃D (jω) =

1 , 1 + jωτ

1 , 1 + (jωτ)α 1 ε̃CD (jω) = , (1 + jωτ)β ε̃CC (jω) =

(15)

0 < α ≤ 1,

(16)

0 < β ≤ 1.

(17)

Even though these models are empirical and do not use fractional derivatives or integrals explicitly, they may be regarded as pioneer applications of FC. Such connection has been addressed by several authors [46, 39, 43, 40, 7, 17, 13].

3 Modeling EIS data In the follow-up the EIS is applied to plant leaves, vegetables, wine, and milk [27–29, 31]. The EIS data are modeled by FO transfer functions and the model parameters are used to characterize the different samples.

Fractional-order modeling of electro-impedance spectroscopy information |

25

3.1 EIS analysis of leaves The study of leaves unveils important characteristics of the whole plant, such as its health condition, its nutrient status, the presence of viruses, and its rooting ability [27]. Leaves can be classified according to the corresponding plant taxonomy, leaf venation, arrangement, or shape, for example. For a given leaf, factors as size, maturation, and age, among others, can also be considered. In this study, leaves of six angiosperms chosen by their shape are treated, namely, the Citrus limon (CL), Ilex aquifolium (IA), Ficus elastica (FE), Hydrangea macrophylla (HM), Acacia dealbata (AD), and Acer pseudoplatanus (AC). In all cases the FO model parameters are correlated with the physical characteristics of the leaves. The diagram of Figure 1 (set-up A) shows the experimental arrangement adopted for EIS measurements [27, 28]. The leaf is submerged in salted water, except its petiole. Two copper electrodes of 0.5 mm diameter and 5 mm length connect the leaf to the measurement circuit. One electrode is inserted into the leaf petiole, aligned with its longitudinal axis. The other electrode is immersed in the salted water. The specimen is connected in series with an adaptation resistance, Rs = 15 kΩ, for signal measurement, while yielding a good signal/noise ratio. A Hewlett Packard/Agilent 33220A function generator applies a sinusoidal 5 V AC voltage to the circuit (i. e., a voltage divider). A Tektronix TDS 2002C two-channel oscilloscope measures the voltages Vab and Vcb . The impedance Z(jω) is obtained for the frequency range 2π × 10 ≤ ω ≤ 2π × 105 rad/s,

Figure 1: Experimental EIS set-up for measuring impedance Z(jω).

26 | A. M. Lopes and J. A. T. Machado at L = 24 logarithmically spaced points, using the expression Z(jω) = Rs ⋅ (

Vab (jω) − 1). Vcb (jω)

(18)

All experiments were conducted with a salted water temperature of 18 °C and during daytime. Several tests demonstrated good stability as regards the oxidation of the copper electrodes, while different electrode geometries revealed a negligible influence on the results. Moreover, experiments with various amplitudes of the excitation signal showed good linearity, allowing data treatment using transfer function concepts. The Bode diagrams of the electrical impedance (Figure 2) are approximated using FO-based models, while minimizing a fitness function, J, based on the Canberra distance [6] between the experimental, Ze , and model, Zm , impedances, i. e., J=

|ℑ[Ze (jωk )] − ℑ[Zm (jωk )]| 1 L |ℜ[Ze (jωk )] − ℜ[Zm (jωk )]| + ), ∑( L k=1 |ℜ[Ze (jωk )]| + |ℜ[Zm (jωk )]| |ℑ[Ze (jωk )]| + |ℑ[Zm (jωk )]|

(19)

where ℜ(⋅) and ℑ(⋅) represent the real and imaginary parts [29, 27]. The function, J, leads to good results because it calculates the ratio between the difference and the sum of two values. Therefore, it is possible to capture the relative error of the adjustment, avoiding “saturation-like” effects, which occur when using the standard Euclidean norm due to the simultaneous presence of large and small values. Several numerical tests proved that the four-parameter model Z(jω) = R +

K 1 + (jω/p)β

(20)

leads to a good approximation to the experimental data. Table 1 summarizes the values of the model parameters that approximate the leave impedance spectra. Figure 3 depicts the normalized {K, p, α} parameters for each leaf. It is observed that three clusters emerge within the representation: {AD, AC}, {FE, HM}, and {IA, CL}.

Table 1: Parameters of the FO-based model for N = 6 leaves. i 1 2 3 4 5 6

Species Citrus limon Ilex aquifolium Ficus elastica Hydrangea macrophylla Acacia dealbata Acer pseudoplatanus

tag

R

K

p

α

J

CL IA FE HM AD AC

3

4

3

0.59 0.72 0.55 0.48 0.37 0.47

0.035 0.288 0.338 0.493 0.398 0.581

8.9 × 10 7.2 × 103 7.5 × 103 2 × 103 7 × 103 1.5 × 104

5.6 × 10 8.4 × 104 7.8 × 104 6.6 × 104 4.9 × 105 5.5 × 105

2 × 10 2.5 × 103 6 × 102 5 × 102 1.2 × 101 1 × 102

Fractional-order modeling of electro-impedance spectroscopy information |

Figure 2: The Bode diagrams of the experimental, Ze , and model, Zm , impedances of leaves.

27

28 | A. M. Lopes and J. A. T. Machado

Figure 3: Locus of the normalized parameters of the FO model for N = 6 leaves.

These clusters are in accordance with the physical characteristics of the leaves. Therefore, it is concluded that incorporating the FO concepts, a simple model could be found to represent leaves.

3.2 EIS analysis of fractal vegetables The EIS is used to measure the electrical impedance, Z(jω), of four “fractal vegetables”, namely, the cauliflower (CF) (brassica oleracea var. Botrytis), broccoli (BR) (brassica oleracea var. italica), round cabbage (RC) (brassica oleracea var. capitata), and Brussels sprout (BS) (brassica oleracea var. gemmifera) [28]. The experimental set-up and the procedure adopted for the EIS measurements are similar to those described in Section 3.2 and are represented in set-up A of Figure 1. Different FO models were adjusted to the measured impedance data, while minimizing function J, as defined in (19). The Havriliak–Negami model describes all vegetables well. However, it is worth noting that the cauliflower and broccoli specimens are close to the Cole–Cole model. Table 2 summarizes the model parameters for the four specimens. The experimental and fitting Bode diagrams are depicted in Figure 4. These results demonstrate that the classical FO empirical formulae constitute simple, yet reliable models to characterize biological structures. Table 2: Parameters of the FO-based model for vegetable fractals. i

Designation

tag

k1

k2

τ

α

β

J

1 2 3 4

Cauliflower Broccoli Round cabbage Brussels sprout

CF BR RC BS

18208.7 1478.8 51627.0 59140.3

496380 404100 109520 784530

18.1 × 10 18.1 × 10−3 6.01 × 10−4 6.01 × 10−4

0.470 0.859 0.444 0.354

1.046 0.897 1.470 1.292

0.0026 0.0027 0.0002 0.0008

−3

Fractional-order modeling of electro-impedance spectroscopy information |

29

Figure 4: The Bode diagrams of the experimental, Ze , and model, Zm , impedances of vegetable fractals.

3.3 EIS analysis of wine In this section the impedance spectra for a set of N = 16 Portuguese wines are determined by means of EIS [29]. The set was chosen for including samples from distinct wine regions [10], involving a mixture of ripe and green, both red and white, styles (Table 3). The experimental set-up for EIS measurements is depicted in set-up B of Figure 1. A parallelepipedic container with dimensions (l × w × h) = (120 × 100 × 55) mm is filled with 200 mL of wine, at room temperature (23 °C). Two 0.5 mm diameter, 15 mm length copper electrodes connect the samples to the measurement circuit. The electrodes are immersed at 5 mm from the bottom of the container and are placed diametrically opposed to each other. An adaptation resistance, Rs = 15 kΩ, is used in series with the specimen under analysis. The electrical impedance is measured in the frequency interval 2π × 10 ≤ ω ≤ 2π × 106 rad/s, at L = 33 logarithmically spaced points.

30 | A. M. Lopes and J. A. T. Machado Table 3: The set of N = 16 wine samples analyzed. i

tag

wine region

wine style

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

W1 W2 W3 W4 W5 W6 W7 W8 R1 R2 R3 R4 R5 R6 R7 R8

Alentejo Alentejo Alentejo Península de Setúbal Tejo Douro Vinhos Verdes Vinhos Verdes Alentejo Alentejo Península de Setúbal Península de Setúbal Bairrada Douro Vinhos Verdes Vinhos Verdes

white white white white white white green white green white red red red red red red green red green red

A good fit occurs for the six-parameter FO model: Z(jω) = K ⋅

(1 +

jω α1 ) z1

⋅ (1 +

(jω)β

jω α2 ) z2

,

(21)

with {K, α1 , z1 , α2 , z2 , β} > 0. Table 4 summarizes the impedance, or electrical, parameters obtained for the N = 16 samples analyzed (to be denoted by the N × S = 16 × 6 matrix ℰ = [e1 , . . . , e6 ]). This table also comprises chemical data, obtained by standard measurement procedures, for four constituents: (i) reducing substances (sugars) – ref: OIV-MA-AS311-01A; (ii) alcoholic strength by volume (alcohol) – ref: OIV-MA-AS312-01A; (iii) total acidity (acidity) – ref: OIV-MA-AS313-01, and (iv) density at 20 °C (by pycnometry) – ref: OIVMA-AS2-01A [18] (to be denoted by the N × T = 16 × 4 matrix 𝒬 = [q1 , . . . , q4 ]). The Bode diagrams of the experimental and approximating model for sample W5 are depicted in Figure 5, revealing the adequacy of expression (21). For the remaining wine samples the results are similar, but they are omitted here for the sake of parsimony. To unveil the relationships between electrical and chemical data the 16×(6+4)-dimensional matrix 𝒫 = [ℰ |𝒬] = [p1 , . . . , p10 ] is defined and the Pearson correlation, rij (i, j = 1, . . . , 10, j > i), between the two vectors of parameters is calculated, i. e., rij =

(pi − p̄ i )T ⋅ (pj − p̄ j ) , ‖pi − p̄ i ‖ ⋅ ‖pj − p̄ j ‖

where p̄ i and p̄ j denote the average values of columns i and j of matrix 𝒫 .

(22)

tag

W1 W2 W3 W4 W5 W6 W7 W8 R1 R2 R3 R4 R5 R6 R7 R8

i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

6700 5000 5800 7000 5500 6000 7500 7500 5500 6800 5000 6000 7500 5000 6500 7200

1000 1000 1300 1000 1000 900 1600 1600 950 1100 1000 1000 1600 1000 1100 1700

0.33 0.32 0.40 0.35 0.32 0.32 0.40 0.40 0.33 0.33 0.33 0.34 0.39 0.32 0.30 0.41

α2 0.88 0.87 0.88 0.86 0.87 0.88 0.75 0.75 0.87 0.92 0.86 0.88 0.75 0.88 0.89 0.88

z2

24 × 104 19 × 104 22 × 104 22 × 104 19 × 104 20 × 104 15 × 104 17 × 104 19 × 104 23 × 104 23 × 104 22 × 104 15 × 104 20 × 104 20 × 104 19 × 104

impedance parameters, ℰ K z1 α1 0.29 0.32 0.31 0.31 0.33 0.29 0.36 0.36 0.32 0.33 0.31 0.33 0.35 0.32 0.35 0.38

β 0.2298 0.2372 0.2678 0.2639 0.2885 0.2888 0.2667 0.2339 0.2859 0.2936 0.2731 0.2941 0.2644 0.2977 0.3651 0.2985

J 12.7 12.7 13.0 12.6 13.1 13.1 9.1 10.0 12.8 13.5 13.4 13.4 12.6 12.6 10.3 10.4

5.47 5.35 5.50 4.31 6.21 6.63 8.01 6.90 5.83 4.93 5.67 5.68 6.77 6.06 8.62 8.11

chemical data, 𝒬 alcohol (alc) acidity (aci) (% v/v) (g tartaric acid/L)

Table 4: Impedance parameters and chemical data of the N = 16 wine samples.

3.0 1.4 1.6 1.0 1.4 1.0 6.4 6.4 1.4 1.9 2.8 2.2 8.0 1.5 2.3 4.3

sugars (sug) (g/L)

988.9 989.0 988.2 988.6 988.8 989.0 1001.0 996.0 991.8 991.0 992.8 992.9 994.8 994.4 996.0 995.7

density (den) (g/L)

Fractional-order modeling of electro-impedance spectroscopy information |

31

32 | A. M. Lopes and J. A. T. Machado

Figure 5: The Bode diagram of the experimental, Ze , and model, Zm , impedances for the wine sample W5 .

Figure 6: Pearson correlation between the parameters used to characterize wine. An arc connects pairs with |rij | ≥ 0.7, and colors represent the magnitude of the correlation.

Figure 6 depicts the correlations by means of a chart, where all pairs with |rij | ≥ 0.7 are connected by an arc and colors represent the magnitude of the correlation. Intra- and inter-model correlations between the parameters are observed. Focusing on the inter-model relationships, strong correlations emerge between alcohol and {z1 }, sugars and {α1 , z1 , α2 }, and density and {β}. In conclusion, the results demonstrate that FO models yield a convincing description and reliable characterization of the wine. Moreover, given the correlations between chemical and electrical parameters, EIS may lead to a simple and straightforward procedure to implement in the winery industry. The impedance, ξ I = {K, β, α1 , z1 , α2 , z2 }, and chemical, ξ C = {alc, aci, sug, den}, parameters characterize wine in a high-dimensional space. Therefore, analyzing all variables simultaneously poses difficulties. In this line of thought, the multi-dimensional scaling (MDS) technique is adopted for reducing dimension and visualizing the relationships between the parameters of the N = 19 samples. The MDS is a visualization method that explores dissimilarities between items [32, 45, 25, 5, 35]. For a set of N items in ℝn , MDS requires a dissimilarity measure δij ,

Fractional-order modeling of electro-impedance spectroscopy information |

33

i, j = 1, 2, . . . , N, and an N × N matrix Δ = [δij ] measuring item-to-item dissimilarities. In classical MDS, matrix Δ is such that δij = δji , δij ≥ 0, i ≠ j, and δii = 0. The MDS represents each item by a point in ℝq , q ≤ n, and positions them in order to obtain a configuration that produces dissimilarities, dij , as close as possible to the original values δij in ℝn . To assess how a configuration of points in ℝq replicates the original values in ℝn the raw stress, ρ, is used. We have 2

ρ = [dij − f (δij )] ,

(23)

where f (⋅) is some kind of monotonic transformation. Adopting dimensions of q = 2, or q = 3, allows for computer visualization of the calculated configuration. One can rotate or translate the set of points in ℝq for a better visualization, since the distances dij remain invariant. The MDS algorithm is fed with two distinct matrices, ΔI and ΔC , corresponding to the EIS and chemical cases. For constructing the matrices the Canberra distance is adopted herein. Therefore, the electrical, ξ I , and the chemical, ξ C , characterization of the wine samples, i, j = 1, . . . , N, is given by [6] S

δijI = ∑

k=1 T

δijC = ∑

k=1

|ξiI (k) − ξjI (k)|

|ξiI (k)| + |ξjI (k)| |ξiC (k) − ξjC (k)|

,

i, j = 1, . . . N,

(24)

,

i, j = 1, . . . N,

(25)

|ξiC (k)| + |ξjC (k)|

where S = 6 and T = 4. The vectors ξiI (k) and ξiC (k) represent the ith sample impedance and chemical parameters, respectively. One must note that using alternative measures in the MDS is a common procedure with different perspectives. By other words, one can choose distinct distances and the corresponding MDS charts to obtain the “best” visualization. Several tests demonstrated that the Canberra distance leads to relevant results. Figure 7 depicts the three-dimensional maps of items for ΔI and ΔC , where each point represents a wine sample. For both measures one observes identical clusters in the MDS, where 𝒞1 = {W2 , W3 , W5 , W6 , R1 , R6 }, 𝒞2 = {W1 , W4 , R2 , R3 , R4 , R7 }, and 𝒞3 = {W7 , W8 , R5 , R8 }. Clusters ℐ1 and ℐ2 differ from 𝒞1 and 𝒞2 only in the items {W3 , R3 }, that change place, while ℐ3 and 𝒞3 remain equal. This means that the MDS is able to identify wine samples with distinct characteristics. In fact, clusters {𝒞1 , ℰ1 } and {𝒞2 , ℐ2 } include ripe wines, of both white and red styles, with the exception of R5 . Cluster 𝒞3 = ℐ3 comprises green wine types, with the exception of R7 . Moreover, by obtaining identical clusters with both chemical and electrical parameters, it is confirmed that the FO models that represent the impedance data from the EIS can be used for wine classification.

34 | A. M. Lopes and J. A. T. Machado

Figure 7: The three-dimensional MDS maps for the N = 16 wine samples and matrices: (a) ΔI ; (b) ΔC .

3.4 EIS analysis of milk The impedance spectra of N = 19 commercial UHT milk varieties are determined by means of EIS [31]. The set comprises samples from distinct brands, with different fat content, and includes a mix of normal, reduced, and fortified milk varieties (Table 5). For the experiments we adopted the arrangement schematized in set-up B of Figure 1. The experimental procedure is identical to the one reported in Section 3.3, and the temperature was 20 °C. Several numerical tests revealed that a good fit between the model and the experimental data is achieved for the six-parameter FO expression (21). Figure 8 depicts the Bode diagram of the experimental and model data, for the M2 milk sample within the bandwidth 2π × 5 ≤ ω ≤ 2π × 106 rad/s. For the other samples the frequency response is of the same type. Tables 6 and 7 summarize the impedance (i. e., electrical) parameters and the chemical values of acidity (aci), density (den) at 20 °C, and protein (pro), fat (fat), sugar (sug), calcium (cal), and dry matter (dma) for the N = 19 samples. These electrical and chemical data will be denoted in the follow-up by the N × S = 19 × 6- and

Fractional-order modeling of electro-impedance spectroscopy information |

35

Table 5: The set of N = 19 milk samples analyzed. i

tag

milk type

1 2 3 4 5 6 7 8 9 10

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

skimmed skimmed skimmed semi-skimmed semi-skimmed semi-skimmed whole whole whole organic semi-skimmed

i

tag

milk type

11 12 13 14 15 16 17 18 19

M11 M12 M13 M14 M15 M16 M17 M18 M19

reduced skimmed fortified skimmed fortified skimmed fortified semi-skimmed fortified (for children) fortified (for children) fortified (for children) fortified (for children) goat semi-skimmed

Table 6: Impedance parameters for the N = 19 milk samples. i

tag

impedance parameters, ℰ K β α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19

33325 36563 45133 29400 35150 34380 59463 35750 31859 32626 27300 34613 30225 28763 27300 30455 20130 9100 28762

0.70 0.70 0.72 0.67 0.73 0.68 0.80 0.69 0.76 0.72 0.64 0.72 0.67 0.70 0.62 0.58 0.56 0.47 0.67

0.68 0.67 0.72 0.67 0.72 0.76 0.72 0.66 0.68 0.76 0.72 0.85 0.77 0.82 0.67 0.65 0.64 0.57 0.72

z1

α2

z2

J

5340 8840 3844 9880 4290 9172 2600 7150 4572 9667 3500 3500 3500 3500 3500 3182 3925 1041 3500

0.57 0.65 0.64 0.59 0.56 0.54 0.68 0.61 0.62 0.57 0.53 0.53 0.53 0.53 0.53 0.48 0.77 0.79 0.53

5606 8233 15813 8906 8190 7717 16225 8450 5999 6656 65000 65000 65000 65000 38750 37264 42418 248300 36875

0.735 0.693 0.710 0.726 0.772 0.690 0.691 0.763 0.710 0.720 0.717 0.747 0.735 0.763 0.791 0.740 0.710 0.731 0.750

N × T = 19 × 7-dimensional matrices ℰ = [e1 , . . . , e6 ] and 𝒬 = [q1 , . . . , q7 ], respectively. The chemical results were obtained using the measurement procedures listed in Table 8. To assess possible correlations between the two quantitative descriptions, that is, between the electrical and the chemical results, one joins ℰ and 𝒬 into the 19 × (6 + 7)-dimensional matrix 𝒫 = [ℰ |𝒬] = [p1 , . . . , p13 ] and the Pearson correlation, rij , i, j = 1, . . . , 13, j > i, between the column vectors is calculated by means of expression (22).

36 | A. M. Lopes and J. A. T. Machado Table 7: Chemical data of the N = 19 milk samples. i tag

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19

chemical data, 𝒬

aci den (cm3 NaOH 1N/dm3 ) (g/dm3 ) 19 18 21 19 18 22 18 18 20 21 17 17 16 17 18 13 18 6 22

1036 1036 1036 1034 1034 1035 1032 1032 1031 1034 1032 1032 1035 1031 1031 1039 1034 1038 1035

pro (g/100 mL)

fat (%)

sug (g/100 mL)

cal (mg/100 mL)

dma (%)

3.5 3.6 4.1 3.5 3.6 4.1 3.2 3.5 3.4 3.8 3.2 3.4 3.3 3.1 3.3 1.9 2.6 1.4 3.8

0.3 0.2 0.3 1.5 1.6 1.8 3.1 3.2 3.6 1.7 0.8 0.9 0.2 1.9 1.7 3.1 2.9 2.7 1.6

5.4 5.4 5.4 5.2 5.2 5.4 4.9 5 5.2 5.5 6.1 5.1 5.1 5 4.4 6.2 5.7 9 5.2

123 123 131 117 120 126 115 122 125 127 114 116 117 137 118 92 95 83 144

9 8.9 9.1 10.1 10.3 10.6 11.9 12.1 12 10.5 9.4 9.7 10.2 10.7 10.6 13.4 12.2 13.8 10.4

Figure 8: The Bode diagram of the experimental, Ze , and model, Zm , impedances for the wine sample M2 .

Figure 9 depicts the correlations between parameters, where all pairs of descriptors with |rij | ≥ 0.7 are connected by an arc and colors represent the magnitude of the correlation. Several intra- and inter-model correlations are observed, where the two descriptions characterizing milk are stronger at parameters {β} ↔ {aci, pro, sug, cal}, {z2 } ↔ {aci, pro, sug, cal}, and {α2 } ↔ {sug}.

Fractional-order modeling of electro-impedance spectroscopy information |

37

Table 8: The procedures adopted for the chemical analysis. component

procedure

1 2 3

acidity density at 20 °C protein

4 5 6 7

fat sugar calcium dry matter

NP 470:1983 NP 473:1983 (pycnometry) analysis of nitrogen by thermal decomposition and chemiluminescence using TOC-VCSN analyzer Röse-Gottlieb gravimetric method adapted (acid hydrolysis) colorimetric method with DNS reagent EDTA complexation (volumetric method) NP 580:1970

Figure 9: Pearson correlation between the parameters used to characterize wine. An arc connects pairs with |rij | ≥ 0.7, and colors represent the magnitude of the correlation.

Since the sets of impedance, ξ I = {K, β, α1 , z1 , α2 , z2 }, and chemical, ξ C = {aci, den, pro, fat, sug, cal, dma}, parameters characterize milk in a high-dimensional space, the MDS is adopted for reducing dimensionality and having a direct visualization. For constructing the matrices ΔI and ΔC corresponding to the EIS and chemical measures, the Canberra distance (24)–(25), with S = 6 and T = 7, is adopted. Figure 10 shows the three-dimensional MDS charts obtained of the N = 19 milk samples, for ξ I and ξ C . In both charts the emergence of three clusters is observed: the elongated group 𝒞1 = {M1 , . . . , M10 } composed of all “normal” and one organic milk, the concentrated set 𝒞2 = {M11 , . . . , M17 }∪{M19 } that includes both reduced and fortified milks plus a goat milk sample, and the single case 𝒞3 = {M18 } formed by a fortified milk suited for children. The results demonstrate that model (21) yields a quantitative description and reliable characterization of milk. Nevertheless, the number of model parameters necessary is high and the adherence between the heuristic model and the experimental data in Figure 8 is limited.

38 | A. M. Lopes and J. A. T. Machado

Figure 10: The three-dimensional MDS charts for the N = 19 milk samples using the Canberra distance (24)–(25) and characterization by means of parameters: (a) impedance, ξ I ; (b) chemical, ξ C .

4 Conclusions In this chapter the EIS was used to determine the electrical impedance spectra of different materials, and FO models were adopted to describe the experimental data. It was shown that FO transfer functions adequately describe the data. Based on the strong correlations found between electrical measurements and “standard” data, it is concluded that impedance obtained from EIS leads to an assertive characterization of the products analyzed. The potential use of simple, non-intrusive, and economical techniques in food production, or in medicine, reveals possible directions to be further explored [51, 36, 44].

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Omar Abu Arqub and Shaher Momani

Numerical solutions of singular time-fractional PDEs Abstract: In this chapter, we propose and analyze a computational algorithm for the numerical solutions of singular time-fractional partial differential equations of Dirichlet function types. By interrupting the n-term of exact solutions, numerical solutions of linear and non-linear singular time-fractional equations of non-homogeneous function type are studied from a mathematical viewpoint. The accuracy properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the presented algorithm and simulated annealing provide a good scheduling methodology to such singular fractional equations. Keywords: Reproducing kernel algorithm, singular, fractional calculus theory PACS: 02.60.-x, 02.40.Xx, 02.30.Jr, 45.10.Hj

1 Introduction Since approximately the year 2000, fractional calculus theory has gained substantial popularity and significance. It has tremendously attractive applications in diverse and widespread fields of physics and engineering, such as viscoelasticity, electromagnetic theory, rheology, diffusive transport, electrical networks, and fluid flow [35, 45, 38, 41, 32]. Fractional-order models are often more accurate than classical integer-order descriptions, because fractional-order derivatives and integrals embed the description of the memory and hereditary properties of different substances. Accordingly, the field of singular time-fractional partial differential equations (PDEs) has attracted interest of researchers in several important phenomena in chemistry, hydrology, fluid mechanic, physics, gas dynamics, and signal processing [17, 40, 29, 21, 43, 20, 34, 12, 24, 23, 39, 37]. This chapter is concerned with a class of singular time-fractional PDEs with various order types of fractional derivatives. More precisely, we consider and use the reAcknowledgement: The authors would like to express their gratitude to the unknown referees for carefully reading the paper and their helpful comments. Omar Abu Arqub, Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan, e-mail: [email protected] Shaher Momani, Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan; and Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia, e-mail: [email protected] https://doi.org/10.1515/9783110571905-003

44 | O. Abu Arqub and S. Momani producing kernel algorithm (RKA) to solve the following class of PDEs: κ1 (x, t)𝜕tαα u(x, t) + κ2 (x, t)𝜕x u(x, t) + κ3 (x, t)𝜕x22 u(x, t) = f (x, t, u(x, t)),

(1)

u(x, 0) = ω(x), { { u(0, t) = υ1 (t), { { u(1, t) = υ2 (t). {

(2)

subject to

Here, we assume that κ1 (x, t) is an analytical real-valued function over the square [0, 1]2 and may take the values κ1 (xλ , tλ ) = 0 for some (xλ , tλ ) ∈ [0, 1]2 which make equation (1) to be singular at (x, t) = (xλ , tλ ). We also assume that 𝜕x = 𝜕/𝜕x, 𝜕x22 = 𝜕2 /𝜕x2 , . . . , and that 𝜕tαα denotes the Caputo derivative of order 0 < α ≤ 1. In this aspect, 𝜕tαα = 𝜕α /𝜕t α and 𝜕tαα u(x, t)

t

1 = ∫(t − τ)−α 𝜕τ u(x, τ)dτ, Γ(1 − α)

0 < τ < t.

(3)

0

Historically, reproducing kernel theory was used first in 1907 to solve harmonic and biharmonic Dirichlet problems [44]. In 1950, it was formalized by knitting it with the reproducing kernel functions [16]. This theory, which is proxy in the RKA, has been used in diverse applications in applied mathematics and engineering modeling [19, 18, 22, 42]. Recently, a broad range of researches have applied the RKA for the solutions of several integral and differential operators alongside their theories [33, 46, 6, 3, 15, 14, 1, 11, 7, 36, 9, 10, 4, 2, 13, 26, 31, 28, 25, 30, 27, 5, 8].

2 Materials and methods A Hilbert space which possesses a reproducing kernel is called an reproducing kernel Hilbert space (RKHS). Through this section, we denote ‖z‖2◼ = ⟨z(∗), z(∗)⟩◼ , where ̂ 1 , W 2 , W 3 }. z ∈ ◼, ∗ ∈ [0, 1], and ◼ ∈ {W21 , W 2 2 2 Definition 1 ([33]). Suppose that z 󸀠 is in L2 [0, 1]. The space W21 [0, 1] is defined as W21 [0, 1] = {z = z(t) : z is an absolutely continuous function on [0, 1]}. Meanwhile, its metric system structure lies in 1

⟨z1 (t), z2 (t)⟩W 1 = z1 (0)z2 (0) + ∫ z1󸀠 (t)z2󸀠 (t)dt. 2

(4)

0

Theorem 1 ([33]). The Hilbert space W21 [0, 1] is a complete reproducing kernel with structure kernel function R{1} s (t) = 1 + min(s, t).

(5)

Numerical solutions of singular time-fractional PDEs | 45

In a similar fashion, if [0, 1] is the domain space in the x-direction, then the space ̂ {1} (x) = ̂ 1 [0, 1] can be defined as ⟨z1 (x), z2 (x)⟩ ̂ 1 = z1 (0)z2 (0) + ∫1 z 󸀠 (x)z 󸀠 (x)dx and R W 2 y 2 W2 0 1 1 + min(x, y). Definition 2 ([46]). Suppose that z 󸀠󸀠 is in L2 [0, 1]. The space W22 [0, 1] is defined as W22 [0, 1] = {z = z(t) : z, z 󸀠 are absolutely continuous functions on [0, 1] and z(0) = 0}. Meanwhile, its metric system structure lies in 1

1

(6)

⟨z1 (x), z2 (x)⟩W 2 = ∑ z1(i) (0)z2(i) (0) + ∫ z1󸀠󸀠 (t)z2󸀠󸀠 (t)dt. 2

i=0

Theorem 2 ([46]). The Hilbert space structure kernel function

0

W22 [0, 1]

is a complete reproducing kernel with

st + 21 st 2 − 61 t 3 , (t) = { R{2} s st + 21 s2 t − 61 s3 ,

t ≤ s,

(7)

t > s.

Definition 3 ([6]). Suppose that z 󸀠󸀠󸀠 is in L2 [0, 1]. The space W23 [0, 1] is defined as W23 [0, 1] = {z = z(x) : z, z 󸀠 , z 󸀠󸀠 are absolutely continuous functions on [0, 1] and z(0) = z(1) = 0}. Meanwhile, its metric system structure lies in 1

1

i=0

0

⟨z1 (x), z2 (x)⟩W 3 = ∑ z1(i) (0)z2(i) (0) + z1 (1)z2 (1) + ∫ z1󸀠󸀠󸀠 (x)z2󸀠󸀠󸀠 (x)dx. 2

(8)

The Hilbert space W23 [0, 1] is a complete reproducing kernel with struc-

Theorem 3 ([6]). ture kernel function

1

120 R{3} y (x) = { 1 120

(Δ1 (x, y) + Δ2 (x, y) + Δ3 (x, y)), (Δ1 (y, x) + Δ2 (y, x) + Δ3 (y, x)),

x ≤ y,

(9)

x > y,

in which Δ1 (x, y) = x2 y2 (126 − x3 − y3 ),

Δ2 (x, y) = y(y(y3 − 10x3 ) − 5x(−24 + y3 )), 3

(10)

3

Δ3 (x, y) = 5xy(y(x − 24) + x(y − 24)).

i+j

i

Henceforth, we denote the following markers: Ω = [0, 1] ⊗ [0, 1], 𝜕xi t j i

j

j

(𝜕 /𝜕x )(𝜕 /𝜕t ), whenever i, j = 1, 2 and and ◼ ∈ {H, W}.

‖u‖2◼

=

= ⟨u(∗, ∘), u(∗, ∘)⟩◼ , where u ∈ ◼, ∗, ∘ ∈ Ω,

Definition 4 ([3]). Suppose that 𝜕x33 𝜕t33 u is in L2 (Ω). The Hilbert space W(Ω) can be defined as W(Ω) = {u = u(x, t) : 𝜕x22 𝜕t22 u is a complete continuous function in Ω and u(x, 0) = u(0, t) = u(1, t) = 0}. Meanwhile, its metric system structure lies in 1

j

j

⟨u1 (x, t), u2 (x, t)⟩W = ∑ ⟨𝜕t j u1 (x, 0), 𝜕t j u2 (x, 0)⟩W 3 j=0

2

46 | O. Abu Arqub and S. Momani 1

1

0

j=0

j

j

+ ∫[ ∑ 𝜕t22 𝜕xj u1 (0, t)𝜕t22 𝜕xj u2 (0, t) + 𝜕t22 u1 (1, t)𝜕t22 u2 (1, t)]dt 1 1

+ ∫ ∫ 𝜕x33 𝜕t22 u1 (x, t)𝜕x33 𝜕t22 u2 (x, t)dxdt.

(11)

0 0

Theorem 4 ([3]). The Hilbert space W(Ω) is a complete reproducing kernel with structure kernel function {2} R(y,s) (x, t) = R{3} y (x)Rs (t).

(12)

Definition 5 ([3]). Suppose that 𝜕x 𝜕t u is in L2 (Ω). The Hilbert space H(Ω) can be defined as H(Ω) = {u = u(x, t) : u is a complete continuous function in Ω}. Meanwhile, its metric system structure lies in ⟨u1 (x, t), u2 (x, t)⟩H = ⟨u1 (x, 0), u2 (x, 0)⟩W ̂1 1

2

1 1

2 2 + ∫ 𝜕t u1 (0, t)𝜕t u2 (0, t)dt + ∫ ∫ 𝜕xt u2 (x, t)dxdt. u1 (x, t)𝜕xt 0

(13)

0 0

Theorem 5 ([3]). The Hilbert space H(Ω) is a complete reproducing kernel with structure kernel function ̂ {1} (x)R{1} (t). r(y,s) (x, t) = R s y

(14)

3 Results Throughout the remainder of the chapter, we will use the following: P = P(x, t, u(x, t)), Pk = P(xk , tk , u(xk , tk )), and Pkn = P(xk , tk , un (xk , tk )) whenever k ≥ 1. To apply the RKA, we must homogenize the non-homogeneous constraint conditions by suitable transformations. This normalizing is needed to put the mentioned conditions into the space W(Ω) to make an agreement feasible between the constructed spaces and the corresponding reproducing kernel functions which are on Ω. For convenience, we still denote the solution of the new equation by u(x, t). So, let κ1 (x, t)𝜕tαα u(x, t) + κ2 (x, t)𝜕x u(x, t) + κ3 (x, t)𝜕x22 u(x, t) = P(x, t, u(x, t)),

(15)

subject to {u(x, 0) = 0, { u(0, t) = 0, { { u(1, t) = 0. {

(16)

Numerical solutions of singular time-fractional PDEs | 47

We define the fractional differential linear operator Π : W(Ω) → H(Ω) as Πu(x, t) := κ1 (x, t)𝜕tαα u(x, t) + κ2 (x, t)𝜕x u(x, t) + κ3 (x, t)𝜕x22 u(x, t).

(17)

Thus, the time-fractional PDE to be solved is governed by the following equivalent functional equation: Πu(x, t) = P(x, t, u(x, t)).

(18)

To build an orthogonal function system of the space W(Ω), we choose a countable dense subset {(xi , ti )}∞ i=1 in Ω, and we define φi (x, t) = r(xi ,ti ) (x, t) and ψi (x, t) = Π∗ φi (x, t), where Π∗ : H(Ω) → W(Ω) is the adjoint operator of Π and is uniquely determined. The normalized orthonormal function system {ψi (x, t)}∞ i=1 of W(Ω) is usually constructed from the process of the Gram–Schmidt orthogonalization of {ψi (x, t)}∞ i=1 as i

ψi (x, t) = ∑ μik ψk (x, t).

(19)

k=1

To apply the RKA, we divide the finite domain Ω into a p × q mesh point with the space step size Δx = p1 in the x-direction of [0, 1] and the time step size Δt = q1 in the t-direction of [0, 1], respectively, in which p and q are positive integers. Anyhow the grid points (xl , tm ) in the space-time domain Ω are defined simultaneously as (xl , tm ) = (lΔx, mΔt),

l = 0, 1, 2, . . . , p, m = 0, 1, 2, . . . , q.

(20)

Algorithm 1. To approximate the solution un (x, t) of u(x, t) on Ω, do the following steps: 1: choose n = pq collocation points in the domain Ω; 2: set ψi (xi , ti ) = Π(y,s) R(x, t)|(y,s)=(xi ,ti ) ; 3: obtain the orthogonalization coefficients μik ; 4: set ψi (xi , ti ) = ∑il=1 μik ψi (xi , ti ), i = 1, 2, . . . , n; 5: choose an initial approximation u0 (x1 , t1 ); 6: set i = 1; 7: set Bi = ∑il=1 μik Pkk−1 ; 8: set ui (xi , ti ) = ∑ik=1 ∑nj=1 Bk ψk (xk , tk ); 9: if i < n, then set i = i + 1 and go to step 7, else stop. Next, we suppose that {(xi , ti )}∞ i=1 is dense on Ω. At first, applying the Schwarz inequality it is easy to see that Π : W(Ω) → H(Ω) is a bounded linear operator, that is, ‖Πu(x, t)‖2W 1 ≤ M‖u‖2W with M > 0. 2

Lemma 1. The system {ψi (x, t)}∞ i=1 is a complete function system in W(Ω) with 󵄨 ψi (x, t) = Π(y,s) R(x, t)󵄨󵄨󵄨(y,s)=(x ,t ) . i i

(21)

48 | O. Abu Arqub and S. Momani Proof. Here, we can see that ψi (x, t) = Π∗ φi (x, t)

= ⟨Π∗ φi (y, s), R(x,t) (y, s)⟩W

= ⟨φi (y, s), Π(y,s) R(x,t) (y, s)⟩H 󵄨 = Π(y,s) R(x,t) (y, s)󵄨󵄨󵄨(y,s)=(x ,t ) i i 󵄨 = Π(y,s) R(y,s) (x, t)󵄨󵄨󵄨(y,s)=(x ,t ) ∈ W(Ω). i i

(22)

Now, for each u ∈ W(Ω), let ⟨u(x, t), ψi (x, t)⟩W = 0, i = 1, 2, . . . . Then, ⟨u(x, t), ψi (x, t)⟩W = ⟨u(x, t), Π∗ φi (x, t)⟩W = ⟨Πu(x, t), φi (t)⟩H = Πu(xi , ti ) = 0. Meanwhile, −1 {(xi , ti )}∞ i=1 is dense on Ω; we must have Πu(x, t) = 0 and from the existence of Π it follows that u = 0. Recall that μik are the orthogonalization coefficients of the orthonormal systems ψij (t) and these bases can be obtained directly from the Gram–Schmidt orthogonalization process of equation (19). Theorem 6. Suppose that Ai = ∑ik=1 μik Pk . If u ∈ W(Ω) is the solution of equations (18) and (16), then ∞

u(x, t) = ∑ Ai ψi (x, t). i=1

(23)

Proof. Since ⟨u(x, t), φi (x, t)⟩W = u(xi , ti ) for each u ∈ W(Ω), while ∑∞ i=1 Ai ψi (x, t) is the , it is convergent in the sense of ‖ ⋅ ‖W . Thus, Fourier series expansion about {ψi (x, t)}∞ i=1 ∞

i

u(x, t) = ∑ ∑ ⟨u(x, t), ψi (x, t)⟩W ψi (x, t) i=1 k=1

i

∞

= ∑⟨u(x, t), ∑ μik ψk (x, t)⟩W ψi (x, t) i=1 ∞

i

k=1

= ∑ ∑ μik ⟨u(x, t), Π∗ φk (x, t)⟩W ψi (x, t) i=1 k=1 ∞

i

= ∑ ∑ μik Πu(xk , tk )ψi (x, t) i=1 k=1 ∞

= ∑ Ai ψi (x, t).

(24)

i=1

For numerical computations, put (x1 , t1 ) = (0, 0). Then from the constraint conditions of equation (16), the value of u(x1 , t1 ) is known. Set u0 (x1 , t1 ) = u(x1 , t1 ) and define the n-term numerical solution of u(x, t) using the truncating version n

un (x, t) = ∑ Ai ψi (x, t). i=1

(25)

Numerical solutions of singular time-fractional PDEs | 49

If W(Ω) is a Hilbert space, then ∑∞ i=1 Ai ψi (x, t) < ∞. Thus, we can guarantee that the numerical solution un (x, t) satisfies the constraint conditions of equation (16).

4 Discussion Next, in the discussion of the numerical computations, some tabulated data and numerical comparisons are presented and discussed quantitatively at some selected grid points on Ω to illustrate the numerical solutions for the following time-fractional PDEs. In the process of computation, all the symbolic and numerical computations are performed by using the Mathematics 9 software package. Example 1. Consider the following linear singular time-fractional PDE: 1 α 1 𝜕t α u(x, t) + xu(x, t) − 𝜕 u(x, t) + 𝜕x22 u(x, s) = g(x, t), t x−t x

(26)

subject to u(x, 0) = 0, { { u(0, t) = tanh(1)t 2α − t α , { { {u(1, t) = 0,

(27)

where 0 ≤ x, t ≤ 1 and 0 < α ≤ 1. Here, the exact solution is u(x, t) = tanh(1 − x)t 2α + (1 − x)t α . Example 2. Consider the following non-linear singular time-fractional PDE: x2 1 𝜕tαα u(x, t) + u3 (x, t) + 𝜕x u(x, s) − 𝜕x22 u(x, s) = g(x, t), sin(x − t) t

(28)

subject to u(x, 0) = 0, { { u(0, t) = 0, { { u(1, t) = 0.25(t 2 + t 3α ), {

(29)

where 0 ≤ x, t ≤ 1 and 0 < α ≤ 1. Here, the exact solution is u(x, t) = 0.25t(t + t 3α−1 ) sin2 (1.5πx). In the previous examples, the function g(x, t) is chosen, so that the given exact solution is satisfied in both the left-hand side and the right-hand side of the mentioned functional equation over the domain Ω. Tables 1 and 2 show the absolute error of approximate solutions of Examples 1 and 2, respectively, obtained at various (x, t) in Ω when α ∈ {0.25, 0.5, 0.75, 1}.

50 | O. Abu Arqub and S. Momani Table 1: Absolute errors of approximating the solution in Example 1. x

t

α = 0.25

α = 0.5

α = 0.75

α=1

0.25

0.25 0.5 0.75 1 0.25 0.5 0.75 1 0.25 0.5 0.75 1

6.21582 × 10 5.94302 × 10−3 6.42683 × 10−3 8.82428 × 10−3 7.69848 × 10−3 1.86470 × 10−3 2.04390 × 10−3 7.24071 × 10−3 3.63447 × 10−3 6.62674 × 10−3 9.58919 × 10−4 3.09686 × 10−4

7.35123 × 10 8.49798 × 10−4 5.92842 × 10−4 1.20461 × 10−5 1.50177 × 10−5 7.33523 × 10−5 5.00177 × 10−5 1.26761 × 10−4 5.92882 × 10−4 8.46588 × 10−5 2.58111 × 10−4 6.21232 × 10−4

6.21582 × 10 9.31519 × 10−5 3.07986 × 10−4 6.45610 × 10−4 5.90189 × 10−5 3.07938 × 10−5 6.64483 × 10−4 8.58409 × 10−4 7.42752 × 10−5 5.94301 × 10−4 2.02546 × 10−5 8.65408 × 10−5

8.01587 × 10−6 1.46653 × 10−6 7.35924 × 10−5 3.56385 × 10−5 6.75667 × 10−6 3.50385 × 10−6 7.24992 × 10−5 1.42784 × 10−5 5.70204 × 10−5 7.46082 × 10−5 2.43103 × 10−5 5.27022 × 10−6

0.5

0.75

−4

−4

−5

Table 2: Absolute errors of approximating the solution in Example 2. x

t

α = 0.25

α = 0.5

α = 0.75

α=1

0.25

0.25 0.5 0.75 1 0.25 0.5 0.75 1 0.25 0.5 0.75 1

2.21144 × 10 3.97297 × 10−3 3.54308 × 10−3 4.80317 × 10−4 2.71112 × 10−4 4.98788 × 10−4 1.95932 × 10−3 9.72555 × 10−3 8.37569 × 10−4 4.95437 × 10−4 1.15031 × 10−3 3.93261 × 10−3

2.75871 × 10 1.98548 × 10−4 3.43027 × 10−5 2.67834 × 10−5 4.59262 × 10−5 7.86321 × 10−4 4.73743 × 10−4 7.52803 × 10−4 4.35716 × 10−4 2.80321 × 10−5 6.19482 × 10−4 1.36539 × 10−4

8.02626 × 10 7.75575 × 10−5 6.14137 × 10−5 4.50001 × 10−5 1.18789 × 10−5 2.35848 × 10−5 6.65337 × 10−5 9.16525 × 10−5 7.65897 × 10−4 9.38907 × 10−5 4.81359 × 10−4 2.54023 × 10−5

1.69753 × 10−5 6.54972 × 10−5 5.06880 × 10−5 1.37874 × 10−5 1.44204 × 10−5 8.18831 × 10−5 4.49813 × 10−5 1.63007 × 10−5 1.74921 × 10−5 7.89076 × 10−5 4.81281 × 10−5 1.07137 × 10−5

0.5

0.75

−4

−4

−4

From these tables, it can be seen that the numerical solutions at each level characteristics α are in good agreement with the exact solutions. Further, the error estimates show that the accuracy of the numerical solutions is closely related to the fill time and fill distance. So, more accurate numerical solutions can be obtained using more mesh points.

5 Conclusions The fundamental significance of the proposed algorithm lies in its ability to handle the major challenges associated with the singular time-fractional PDEs in terms of

Numerical solutions of singular time-fractional PDEs | 51

high non-linearity, non-homogeneity, fractional level characteristics, and the notion that the nature of Dirichlet conditions may appear. For the sake of clarity, in this chapter the solutions were represented in the form of series in the extended inner product spaces which showed higher accuracy in numerical computations. The comparative studies based on the absolute natural error function sense show that the RKA approximate values are more acceptable in terms of accuracy and stability. As a final remark, the suggested method can be easily applied to much more complicated non-linear singular fractional PDEs.

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Clara M. Ionescu, Dana Copot, and Cristina Muresan

A multi-scale model of nociception pathways and pain mechanisms Abstract: To develop suitable pain management policies and drug delivery assist devices for analgesia (i. e., pain alleviation), it is necessary to have a mathematical model which captures the essential dynamics of this complex process. Recent work points to the fact that pain can be characterized by several dynamic stages, including anomalous diffusion and spatio-temporal dependency on tissue characteristics. This chapter presents a physiologically based mathematical framework to capture nociceptor pathways and pain reception, transmission and perception, in the human body. The main difference with previous studies is the explicit incorporation of fractional calculus tools as a natural way to characterize biological phenomena. Next, we observe the effects in skin impedance in the presence of nociceptor stimulation. For this purpose, a prototype device has been carefully designed to allow for the application of a noninvasive measurement protocol. Bio-electrical skin impedance captures the changes in tissue content at various time instants, sensor locations, and stimulus trains. The existence of a memory effect – or residual pain – is observed from the data. Keywords: Nociceptor pathway, chronic pain, analgesia, bio-impedance, fractionalorder impedance models, non-invasive measurement, mathematical model, residual pain, memory MSC 2010: 65D30, 92C30, 92C50, 93A30, 93B30, 93C10, 93C80, 93C95, 97M10

1 Introduction Pain is rather a subjective and personal sensation, especially in awake and aware individuals [26, 40]. The self-evaluation metrics often become biased by the tissue mem-

ory, i. e., perception of pain in its absence, or artificially elevated levels of pain due to Acknowledgement: This chapter has been financially supported by Flanders Research Center, grant nr. G026514N and G008113N, and post-doctoral fellowship grant nr. 12B3415N (C. M. Ionescu). Part of this work has been carried out within COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology). Clara M. Ionescu, Dana Copot, Department of Electrical energy, Metals, Mechanical Constructions and Systems, Research group on Dynamical Systems and Control, Ghent University, Technologiepark 914, 9052 Ghent Zwijnaarde, Belgium, e-mails: [email protected], [email protected] Cristina Muresan, Department of Automation, Technical University of Cluj Napoca, Memorandumului nr 28, Cluj, Romania, e-mail: [email protected] https://doi.org/10.1515/9783110571905-004

56 | C. M. Ionescu et al. anxiety, discomfort, and fear [7]. The most frequently used tools to assess acute pain are the numeric rating scale (NRS) and the visual analog scale (VAS), ranging from 0 (no pain) to 10 (excessive pain), e. g., by means of the Wong–Baker faces scale. However, many critically ill patients are unable to communicate effectively because of cognitive impairment, sedation, paralysis, or unconsciousness (e. g., due to general anesthesia). Another group unable to communicate pain are neonates and infants [12]. As such, no single tool is universally accepted for use in the noncommunicative (anesthetized) patient [20, 16]. When a patient cannot express himself, observable indicators – both physiologic and behavioral – have been treated as pain-related indicators to evaluate the pain level [14]. Thus the numbers are simply estimates of the perception of the pain, based on past personal experience of the caregiver. The state of absence of pain due to medication is referred to as analgesia. It is important to admit that patient analgesic needs can differ depending on clinical circumstances, and that for any given patient therapeutic targets are likely to change over time, mainly due to drug trapping [4]. Thus, achieving patient comfort and ensuring patient safety, including avoidance of over- and under-dosage, relies on accurately measuring pain, agitation, sedation, and other related variables. This should be evaluated with validated tools that are easy to use, precise, accurate, and sufficiently robust to include a wide range of behaviors. From the point view of analgesia and chronic pain management, the community is still missing an adequate pain measurement tool based on objective processing of information. A comprehensive review of available tools to extrapolate on pain levels is given in [2]. A linear input-output-based model was identified by performing thermal cold stimuli into dental nerves and measuring the resulting electrical activity correlated to pain [9]. The model was a simple second-order transfer function with damping factor and impulse response corresponding to measured electrical activity in interdental nerves. This crude model was further improved to better approximate the intra-patient variability and plasticity of pain sensation after repeated stimuli [8]. Further in vivo tests indicated that modulation is present in the electrical activity when pain is perceived by the subject, suggesting thus that a frequency dependence is necessary. Non-linear terms in sine and cosine functions have been introduced in [10] to predict this non-linear effect. Somewhat later, a review of multi-scale processes involved in nociception and pain sensation has been made, summarizing all steps from thermal stimuli [45]. Although the review provides an excellent overview, it concludes that the mechanistic processes are far from being well understood and that engineering tools need to be further employed for delivering useful models for assessing pain in humans. A model for electrical activity aroused from thermal nociceptor detection and transmission at the neuronal level is then given. Later studies on thermal pain indicated the presence of adaptability and variability in pain sensation as a result of the noxious stimulus intensity degree and the pattern of stimulation [27]. In engineering terms, this

A multi-scale model of nociception pathways and pain mechanisms | 57

is due to variability of disturbance profiles (i. e., stimuli) and thus the excitatory input to the measured response variation provides a spatio-temporal change in the pathway. Simple linear models of classical system engineering theory can no longer capture such changes without increasing the complexity of the problem formulation.

2 Physiological background The detection of stimuli that are capable of producing tissue injury is termed nociception. These primary sensory neurons have cell bodies in the dorsal root ganglia or in the trigeminal ganglia and possess naked peripheral endings that terminate in the skin, mostly in the epidermis (upper layer of the skin) [33]. Nociceptors, the receptors of pain, are the first unit in the series of neurons related to nociceptive pain. They transduce mechanical, chemical, and thermal energy into ionic current (noxious stimuli result in depolarizations that generate action potentials), conduct the action potentials from the peripheral sensory neurons to the central nervous system (CNS), and convert the action potentials into neurotransmitter release at the presynaptic terminal [33]. In peripheral nerves, nociceptors have unmyelinated (C-fibres) or thinly myelinated (Aδ-fibres) axons [25]. Nociceptors have a lower conduction velocity compared to other peripheral sensory nerve fibres. Generally, the Aδ-fibres have a medium diameter of 2–6 μm with a conduction velocity of 12–30 m/s. In comparison, the Cfibres have a small diameter of 0.4–1.2 μm with a conduction velocity of 0.5–2 m/s. These types of fibres account for the fast and slow pain responses, respectively. Of another type of fibres, the large-diameter Aβ-fibres, the conduction velocity is about 30–100 m/s. According to the response to different stimuli, nociceptors can be further classified as high-threshold mechanoreceptors, chemoreceptors, temperature-sensitive receptors (heat/cold), polymodal nociceptors, and mechano-insensitive (silent) nociceptors. About 70 % of the Aδ-fibre nociceptors are mechanical, 20 % are mechano-heat, and 10 % are mechanic-cold nociceptors [25]. Ion channels in the plasma membrane of nociceptors have a key role in the transduction of stimuli; these are proteins located in the cell membrane that selectively mediate the transmembrane transportation of specific ions or molecules. The ion channels include heat activated channels, capsaicin receptor-dependent channels, adenosine triphosphate (ATP)-gated channels, proton-gated channels, nociceptor-specific voltage-gated NA+ channels, and mechano-sensitive channels. All these types of channels are essentially converted from closed to open states by mainly three types of stimulus: thermal (threshold 43 °C), mechanical (threshold 0.2 MPa), and chemical. The voltage-gated channels are the most important. These

58 | C. M. Ionescu et al. respond to membrane depolarization or hyperpolarization and are substantial to the generation and transmission of electrical signals along axons. When a noxious stimulus reaches a nociceptor, the corresponding ion channels will be opened, which will induce a transmembrane current and increase the membrane voltage. When this voltage increases to the threshold, specified sodium channels will open in a positive feedback mode that results in the depolarization of the membrane, eventually generating an action potential. Primary afferent nociceptors mostly terminate in the spinal cord, which has an important role in the integration and modulation of pain-related signals. Second-order neurons receiving input from nociceptors and projecting to the brain are located in both superficial and deep laminae of the dorsal horn [25]. These cells often have convergent inputs from different sensory fibre types and different tissues. Both pre- and post-synaptic elements are strongly gated by descending excitatory and inhibitory influences from the brain. The inhibitory influences use neurotransmitters that are mimicked by some analgesic drugs. During consciousness, using MRI, it is possible to identify those brain areas directly related to pain [25, 24]. Such a stimulus reliably leads to activation of multiple brain areas, jointly termed the pain matrix. Different areas represent different aspects of pain. The primary and secondary somatosensory cortices are activated to discriminate the location and intensity of a painful stimulus. The anterior cingulate cortex, frontal cortex, and anterior insula regions may be related to the cognitive and emotional components. The problem is that these areas show significant modulation depending on the context of the stimulus, e. g., degree of attention, anxiety, expectation, depression, and analgesic drug treatment. There is an established relation between the nociceptor pathway and dynamics of potassium channels, i. e., the sodium–potassium pump, for signaling between intracellular fluid and extra-cellular fluid (ECF) in the biological tissue [33]. The observed increase in potassium concentration in the ECF varies between 0.1 and 10.0 mmol/L and depends on stimulation frequency, intensity, and duration [25]. In vitro validation studies have been performed to verify the use of the proposed models for detecting changes in the concentration of these cations in controlled environment solutions [3]. From this initial step, we extrapolated that one may measure non-invasively the changes in the signaling pathways, by means of bio-electrical impedance, via the skin [43]. The proposed method for measurement is based on sending an excitatory electrical signal to the skin, while measuring its response as voltage and current changes. By changing the signaling conditions (i. e., with mechanical nociceptor stimulation) the impedance so measured changes its values as well, as a result of changes in the composition of the intra-cellular fluid and the ECF by the movement of the cations.

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Figure 1: Schematic overview of main sequences in nociceptor pathways and model rationale equivalent.

3 Multi-scale model development Originating from our prior work on modelling biological tissue with fractional-order impedance models (FOIMs), the following extension is proposed. An overview of the processes and equivalence to model development is depicted in Figure 1, including changes at the molecular level. The physiological pathway of pain can be described as four main processes [33]: – transduction – when a stimulus is applied to the skin, the nociceptors located there trigger action potentials by converting the physical energy from a noxious thermal, mechanical, or chemical stimulus into electrochemical energy; – transmission – the signals are subsequently transmitted in the form of action potentials (similar to pulse trains) via nerve fibers from the site of transduction (periphery) to the dorsal root ganglion or the trigeminal ganglion, which then activates the interneuron; – perception – the appreciation of signals arriving in specified areas in the cerebral cortex as pain; and – modulation – descending inhibitory and excitatory input from the brainstem that influences (modulates) nociceptive transmission from the spinal cord. The stimulatory effects of nociception are essentially considered an ultra-capacitor, which is represented by a non-rational form of a transfer function model in (jω)n , with n being any real number [31, 11]. Specifically, the skin–electrode interface, the

60 | C. M. Ionescu et al. stratum corneum, and ionic pathways can be modeled as elements in an electrical network. Various models describe this interface using constant or current-dependent resistive-capacitive equivalent circuits [39, 32, 15, 13]. Using fraction expansion theory, a lumped FOIM can be obtained as a fractional-order integral [19]. Similarly, transmission in signaling pathways occurs via neuronal activity, already modeled with FOIMs from resistance-inductance equivalent electrical network elements [17], expressed by a fractional-order derivative. The perception model based on combined exponential and power law functions seems to be a good candidate for capturing essential electrical activity modulated in the brain [1]. Plasticity in synaptic variance is introduced in a layer-based sensory area in the cortex by reverse node engineering modeling [37]. In the case of pain perception, the combined effect can be obtained by using the Mittag-Leffler function, which is well known to capture hybrid exponential and power law behavior in biological tissues [24, 41]. Diffusion of perception sensory activity in the brain using the Mittag-Leffler function in the time domain corresponds to a non-integer-order derivative easily expressed in the frequency domain [44]. Layered activity can be represented by ladder networks with serial connection of RC cells. To account for plasticity, the RC cells are not identical; instead they behave as a memristor with unbalanced dynamics. For instance, it is expected that the first pain perception is more intense than the second, given the latency of the delayed pain stimulus (i. e., sharp first increase followed by slowly decaying tail). Assuming the brain cortex area to be a porous tissue whose porosity varies (i. e., intra- and extra-cellular space tissue with different densities), one can model the changes in viscosity as a function of this porous density. It has been shown that fractional-order derivatives are natural solutions to anomalous diffusion equations [19, 24, 44, 28]. The use and physical interpretation of this very useful fractional calculus tool has been discussed in several works, e. g., [24, 30, 21, 22, 18]. The net advantage of using the Mittag-Leffler function is that it allows for the introduction of memory formalism [38], therefore taking into account the tissue rheology. The mixed area in brain tissue will introduce a dynamic viscosity and thus a dynamic perception of nociceptor-induced pain [42, 23]. Finally, the perception and modulation activity can be characterized yet again by an FOIM as differ-integral (depending on the sign of the non-rational order) [44, 36]. In conclusion, a lumped FOIM comprising the main processes described above is given by ZFOIM (s) = R +

TD TS + + Psα3 , sα1 sα2

(1)

where α1 , α2 , α3 ∈ (−1, 0)∪(0, 1) and TD denotes transduction, TS denotes transmission, and P denotes perception. A calibration factor has been added, a gain R. It may be that not all terms in this model are necessary at all times, as some of the physiological

A multi-scale model of nociception pathways and pain mechanisms | 61

processes may be impaired in some applications (e. g., analgesia will fix the effect of the perception term in P at zero) [5]. The units are arbitrary, as the model is defined as a difference to the initial state of the patient – due to the use of fractional derivatives – and not as absolute values. This enables patient specificity since no generic model is assumed to be valid and thus broadcasts a new light upon the interpretation of such models.

4 Preliminaries Figure 2 depicts the flowchart of the measurement protocol. For this purpose, a prototype device has been developed, ANSPEC-PRO, depicted in Figure 3, and the electrode is placed in the hand palm.

Figure 2: Flowchart of the measurement process.

Figure 3: Left: The ANSPEC-PRO prototype for non-invasive measurement of bio-electrical skin impedance. Right: Placement of the electrodes during proof-of-concept measurements; two current carrying electrodes (white, red) and one pick-up electrode (black).

62 | C. M. Ionescu et al. The measurement flowchart can be summarized as follows: – design a multi-sine signal with 29 components in the frequency interval 100– 1500 Hz, with step interval of 50 Hz; the multi-sine signal is sent with an amplitude of 0.2 mA, a factor 5 below the maximum allowed by clinical standards [6]; – send this signal and acquire the measured signals at sampling frequency of 15 KHz; – use a 3M three-point electrode sensor in the hand palm (CE-marked) (according to MDD93/42/EEC); – measure current and voltage via a National Instruments (Texas, USA) device (cRIO9074 with NI9201- and NI9263-slots); – store the signals online or on the computer for further processing. The computer is a laptop with the operating system Windows 7 Enterprise 64-bit and an INTEL(R) Core(TM) i7-6600U [email protected] processor. A graphical user interface allows monitoring of signal quality. The measurement requires three-point electrodes: two current carrying electrodes and one pick-up electrode. The latter measures the voltage without carrying any currents; hence, no polarization occurs. All electrodes were placed on the palmar side of the hand (see Figure 3). A calibration of the measurements was performed for each volunteer by measuring for 10 minutes without nociceptor stimulus applied and without removing the electrodes. The study was carried out on one healthy volunteer, without pain relief medication treatment at the measurement moment. In this individual, two consecutive measurements were executed to investigate the repeatability and existence of pain memory. Sensors were placed on the right hand and the nociceptor stimulation was applied at the same location (i. e., the same hand). The protocol summarized in Table 1 was applied. Table 1: The time intervals and actions within the 10-minute measurement protocol. The P/NP denote the acronym used in the figures to indicate the case. Time interval (min)

Nociceptor stimulation

0–2 2–3 3–6 6–7 7–10

Absent (NP1) Present (P1) Absent (NP2) Present (P2) Absent (NP3)

The measured signals are filtered for noise prior to the application of non-parametric identification methods [29]. Given the input is of sinusoidal type (A sin(ωt)), the

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impedance is a frequency-dependent complex variable evaluated as Z(jω) =

SXX (jω) , SXY (jω)

(2)

where SXX (jω) denotes the auto-correlation spectrum of the signal, SXY (jω) denotes the cross-correlation spectra between the input-output signals, ω = 2πf is the angular frequency in rad/s, with f being the frequency in Hz, and j = √−1. The classical periodogram filtering technique has been applied with no overlapping interval, with windowing function Blackman implemented in the Matlab environment [29]. The impedance is then evaluated every minute from online data streaming and plotted against frequency. This is then a frequency response either in complex form (real and imaginary parts), or in Bode plot form (magnitude and phase).

5 Results and discussion The time-based current and voltage signals were acquired at a sampling frequency of 15 kHz. A snapshot of a small interval is depicted in Figure 4. In this figure one observes the input signal (current) remains the same at all times, while the recorded output signals (skin response) undergoes changes between the NP and P intervals (recall the protocol from Table 1).

Figure 4: Time-based input-output signals for a snapshot of the interval with absent (NP)/present (P) nociceptor stimulation.

The frequency response of the complex impedance calculated using (2) and illustrated by means of changes with respect to calibrated impedance prior to the test is depicted in Figure 5 for one individual test. In this figure one may observe the following: – applying the same mechanical nociceptor stimuli, the real part of the impedance decreases from P1 to P2, i. e., the level of perception of the pain is lower;

64 | C. M. Ionescu et al.

Figure 5: Changes in impedance as a function of frequency by means of its real and imaginary parts, calculated per interval of absent (NP)/present (P) nociceptor stimulation.

–

the impedance values during absence of nociceptor stimulation decrease from one interval NP1 to another, i. e., NP2, NP3, while NP2 overlaps with P1, i. e., the memory of the stimulation persists in the tissue.

For the same individual, for the pain P1 interval, the fitting of the FOIM from (1) onto the frequency response complex impedance data is depicted in Figure 6. The fitting was again obtained using non-linear least squares identification, with steepest gradient descent, in an iterative manner. Iteration was performed to avoid local minima and

Figure 6: Identified FOIM for the raw impedance data P1.

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the number of iterations between the identified results varied between #2 and #4 in all data. The iteration was stopped when the model parameters changed less than 5 %. It is important to understand that the method and models developed here are uniquely defined for each individual. In other words, the reporting of the model values is not relevant here because the data are expressed with respect to the initial moment of measurement, whereas the state of the patient is taken as a reference. Hence, all values reported are in fact calibrated for that reference value of impedance and each individual has his/her own initial state values. The use of FOIMs is now justified by the data in some sense that, indeed, tissue memory exists and it is a feature naturally explained with properties of mathematical models from fractional calculus. The detailed description of properties of FOIMs has been given in numerous other reports, hence it is omitted here [24, 30]. The data are nevertheless relevant, for they support a method, a device, and mathematical models to provide an indication of change in bio-electrical impedance measured via skin electrodes correlated with absence/presence of nociceptor stimulation. This is a first step towards developing a full measurement set-up and an algorithm for quantifying related pain levels. Our proposed tools are in the same line of thought as those presented in [34, 35]. An intelligent analysis system based on fuzzy logic models was successfully tested in post-operative patients, whereas patient-controlled analgesia (morphine-based) was titrated from the determined index. With respect to their work, our work differs in that it delivers a mathematical framework related to the actual tissue dynamics (i. e., memory, dielectric) properties and thus justifies the use of FOIMs. The changes in the skin impedance affect both time and frequency domains, as suggested by our results reported in this chapter. These changes are evaluated with respect to an initial state of the individual, e. g., when it experiences no pain or when the level of pain is already characterized via other assessment tools (verbal or non-verbal, depending on the state of the patient). It should be noted that in certain situations, care must be taken when referencing to other states of the patient. For instance, if the painless state is recorded before a surgical intervention involving general anesthesia of the patient, the composition of the interstitial tissue will be greatly affected by the cocktail of medication given during this intervention. Following ICU evaluation, pain levels will then have to be referenced to a more recent state of the patient. However, if the pain assessment is to be performed during the surgical nociceptor stimulation, then the referencing with the pre-operatory state of the patient may be relevant. The present study is limited in the number of individuals measured. No actual chronic pain patients or ICU post-operatory pain patients have been included. A correlation to clinical practice indices, such as the Wong–Baker faces scale, should be investigated using a larger population in order to extract a mathematical relationship between model parameters and clinical levels of pain. Although the method is personalized, i. e., the values are calibrated to the initial state of the individual/patient, an analysis of the influence of BMI on the accuracy of the estimators should also be

66 | C. M. Ionescu et al. performed. We do not claim the values given here are reference values. Rather, they are specific for the individual included in this study.

6 Conclusions This chapter introduced a physiological and mathematical framework to allow understanding the pain mechanism and detect the nociception stimulation effects in skin impedance in a healthy volunteer as a proof of concept. The notoriously successful FOIM formulation has proven once more useful to characterize time and frequency evolution of a pre-defined protocol of nociceptor stimulation applied non-invasively in one subject. The next steps are an in-depth analysis in post-operatory patients under pain alleviation treatment and correlations to standard clinical practice of pain level assessments.

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Joana Pinheiro Neto, Duarte Valério, and Susana Vinga

Variable-order derivatives and bone remodeling in the presence of metastases Abstract: Bone tissue is not static: it is constantly being remodeled in a dynamic equilibrium between bone cells. The main types of cells involved are the osteoclasts, responsible for bone resorption, and the osteoblasts, which rebuild it. Should the bone be affected by metastases, these modify the dynamics of the remodeling process to their own profit. They may result either in an osteolytic tumor (in which the bone gets too thin) or in an osteoblastic one (in which bone growth is excessive and inordinate). Existing treatments attack the tumor directly, but also try to restore the equilibrium associated to healthy bone. A set of partial differential equations can be used to describe the dynamics of this system. Since diffusion processes are involved, and anomalous diffusion can often be found in biological processes, these equations can be considered of fractional order. Furthermore, the interference of an osteolytic metastatic bone disease can be modeled as a variation in time of the differentiation order: in particular, to a reduction, corresponding to subdiffusion phenomena. In this variable-order model, treatments will correspond to a recovery of the original differentiation order. Keywords: variable-order derivatives, bone remodeling, tumor, bone metastasis, anomalous diffusion MSC 2010: Primary 26A33, Secondary 34A08, 34K37, 35R11, 60G22, 35Q92, 92C37

1 Introduction Fractional calculus is an appropriate mathematical tool to describe the non-linear relationship with time of anomalous diffusion [24]. Of course, given that many biological processes often present anomalous diffusion, fractional calculus can be used to describe the related phenomena (e. g., subthreshold nerve conduction, viscoelasticity, bio-electrodes). But several physical processes also appear to exhibit a Acknowledgement: This work was supported by FCT, through IDMEC, under LAETA, projects UID/EMS/50022/2013, BoneSys, PERSEIDS (PTDC/EMS-SIS/0642/2014), and IF/00653/2012, and also under INESC-ID, project UID/CEC/50021/2013. Joana Pinheiro Neto, Duarte Valério, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, Lisbon, 1049-001 Portugal, e-mails: [email protected], [email protected] Susana Vinga, IDMEC and INESC-ID, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, Lisbon, 1049-001 Portugal, e-mail: [email protected] https://doi.org/10.1515/9783110571905-005

70 | J. P. Neto et al. fractional-order behavior varying with time or space [22]. In the field of viscoelasticity of certain materials, the temperature effect in small-amplitude strains is known to induce changes from an elastic to viscoelastic/viscous behavior, where real applications may require a time-varying temperature to be analyzed. The relaxation processes and reaction kinetics of proteins, described by fractional differential equations, have an order that depends on the temperature. The behavior of some diffusion processes in response to temperature changes can be better described using variable-order elements rather than time-varying coefficients, among other cases [22]. Thus, fractional-order integrals and derivatives with variable orders have been introduced; the order can change either as a function of time t or of some other variable x [22, 47]. Bone is constantly being renewed, i. e., destroyed and formed again, due to cells termed osteoclasts and osteoblasts, respectively. In the adult skeleton, both processes are in balance and tightly coupled through autocrine and paracrine factors between bone cells that allow maintaining a constant bone density. This specific micro-environment provides the necessary conditions for the growth and proliferation of tumor cells, and so bone is a common site for the development of metastases (in particular from primary breast and prostate cancer). It is possible to replicate the bone remodeling process using mathematical and computational models with differential equations to represent the control mechanisms involved [19]. These models have been extended to include the effects of tumor disruptive pathologies in the bone dynamics, as metastases contribute to the decoupling between bone resorption and formation and to the self-perpetuating tumor growth cycle [2]. The effects of currently used therapies were also contemplated [2], through the pharmacokinetic (PK) and pharmacodynamic (PD) combination of anti-cancer (chemotherapy) with anti-resorptive treatments (bisphosphonates or monoclonal antibodies) [11–13]. Fractional- and variable-order derivatives can be successfully used in modeling the dynamics of bone remodeling. These models are here revisited and further studied. The remainder of the chapter is organized as follows. Bone remodeling physiology, PK/PD concepts, and published integer mathematical models are presented in Section 2. Fractional- and variable-order models are addressed in Section 3. Finally, results are discussed in Section 4. For fractional- and variable-order concepts and definitions, see [42]. All models presented here use dimensionless variables and parameters, including the cell populations, except when explicitly said otherwise in Tables 2, 3, and 4. The Grünwald– Letnikoff definition of fractional derivatives is employed when the order α is constant; Dα(t) or Dα(t,x) refers to the Grünwald–Letnikoff type-𝒟 variable-order derivative, 𝒟 α(t) D−∞+ or 𝒟 Dα(t,x) −∞+ , respectively.

Variable-order derivatives and bone remodeling in the presence of metastases | 71

2 Bone remodeling This section begins by introducing bone physiology concepts, in Section 2.1. PK and PD models follow, in Section 2.2. Finally, a road map through published integer models that reflect bone dynamics is presented in Section 2.3.

2.1 Introducing bone physiology The skeleton supports and protects vital organs and is an active metabolic tissue [10]. It is not static: it constantly undergoes remodeling. This process is spatially heterogeneous, with regular but asynchronous cycles that can take place on 5 % to 25 % of the total bone surface available [14]. About 10 % of the bone is renewed each year [21]. This reconstruction occurs within a basic multi-cellular unit (BMU), a temporary anatomical structure where bone is resorbed by cells termed osteoclasts, and sequentially formed due to cells called osteoblasts [29]. An active BMU comprises 10–20 osteoclasts that remove old and damaged tissue, and around 1000–2000 osteoblasts that secrete and deposit unmineralized bone matrix, directing its formation and mineralization into mature lamellar bone [37]. The BMU can be seen as a mediator mechanism, as it bridges individual cellular activity to whole bone morphology [35]. This process is balanced by a tightly controlled mechanism between the different clusters of bone cells involved, acting through autocrine and paracrine factors between them [35]. Osteoclasts are multi-nucleated cells resulting from mono-nucleated hematopoietic stem cells fused with progenitors cells. As they both express Receptor Activator of Nuclear Factor kB (RANK) and Macrophage Colony-stimulating Factor Receptor (cfms), by biding to RANK-ligand (RANKL) and Colony-stimulating Factor 1 (CSF-1), respectively, these cells differentiate into active osteoclasts capable of bone resorption. Osteoprotegerin (OPG), a soluble decoy receptor for RANKL and a physiological negative regulator of osteoclastogenesis, also plays an important role in osteoclast performance [5, 35]. It is their generation rate that dictates the BMU extension, whereas their life span determines the depth of the resorption [3]. Osteoblasts are mono-nucleated cells that differentiate from mesenchymal stem cells (MSCs). They are controlled by bone morphogenetic protein (BMP), Wnt signaling, vitamin D, and other factors. Parathyroid hormone (PTH) receptors in the osteoblasts upregulate the expression of RANKL, in the presence of the hormone, which binds to RANK expressed in osteoclast precursors. Thus, PTH promotes their activation and bone resorption. These cells also produce OPG, which inhibits osteoclastogenesis by binding to RANKL. The secretion of OPG is reduced in response to PTH, which contributes further to osteoclastogenesis. Osteoblasts can undergo apoptosis or differentiate into osteocytes or into bone lining cells [14, 36].

72 | J. P. Neto et al. Bone resorption and formation is mainly regulated by the RANK/RANKL/OPG pathway and PTH. It is activated by mechanical stimuli on the bone or due to systemic changes in homeostasis which result in the production of estrogen or PTH [35]. The latter is triggered in response to a reduced calcium concentration, which leads to calcium release from the bone matrix, and inhibited when an elevated calcium concentration is sensed [43]. The remodeling process begins when PTH triggers two mechanisms in the osteoblasts. In one of them, the existing PTH reduces the osteoblast secretion of OPG that, by being a soluble decoy receptor for RANKL and allowing it to consequently bind to RANK, promotes osteoclastogenesis. In the other, PTH receptors also upregulate the expression of RANKL, which again binds to RANK, further promoting osteoclast activation and bone resorption [35]. The initiation of the formation phase is coupled to the resorption phase in a process not yet fully understood, as factors released from the bone matrix during resorption (Insulin Growth Factors I and II [IGF-I, IGF-II] and Transforming Growth Factor-β [TGF-β]) may be involved in this coupling. Bone formation takes place even in the presence of malfunctioning osteoclasts, which has led to the hypothesis that osteoclasts produce the coupling factors responsible for attracting and regulating osteoblasts to the sites of bone resorption [5]. At the resorbed site, osteoblasts commence bone formation and replace the resorbed bone by the same amount, ending the bone remodeling cycle [27]. See Figure 1. Metastatic cancer cells accelerate the remodeling process and disturb the balance between bone cells, disrupting its biochemical regulation [21] and destroying bone integrity. Cancer metastases usually appear where bone remodeling rates are high: the pelvis, the axial skeleton, or bones with abundant bone marrow [5, 41]. Bone metastases can be osteolytic (increased bone resorption) or osteoblastic (bone formation is stimulated in an unstructured way). Both are present in any case, although out of balance. Metastases of breast cancer are prone to develop osteolytic metastasis; those of prostate cancer, osteoblastic metastasis [45]. Tumor cells in osteolytic metastases stimulate osteoclast activity and receive positive feedback from factors released by the bone micro-environment during bone destruction [7, 10]. TGF-β released from the bone matrix during resorption stimulates tumor growth and PTH-related protein (PTHrP) production in metastatic cells. By binding to PTH receptors on cells of osteoblastic lineage, RANKL levels are then enhanced. Subsequently, osteoclasts are activated, leading to increased bone resorption [7]. Osteoclast activity results in the release of TGF-β from degraded bone, further stimulating tumor growth and PTHrP secretion. Thus a vicious cycle arises [27]. Tumorous cells of osteoblastic metastases grow as bone expresses endothelin-1 (ET-1). ET-1 stimulates osteoblasts through the endothelin A receptor (ETR), activating Wnt-signaling. Tumor-derived proteases contribute to the release of osteoblastic factors from the extracellular matrix, including TGF-β and IGF-I. RANKL levels are increased due to tumor-induced osteoblast activity, leading to the release of PTH and promoting osteoclast activity [7]. Thus, the tumor micro-environment leads to the accumulation of newly formed bone.

Variable-order derivatives and bone remodeling in the presence of metastases | 73

Figure 1: Biochemical processes of healthy and abnormal bone remodeling. For a healthy bone environment (A), PTH stimulates RANKL production by osteoblasts as the RANK/RANKL/OPG pathway plays an important role in bone resorption and formation. For the progression of osteolytic bone metastases (B), the vicious cycle is due to bone-derived tumor growth factors (IGFs, TGF-β, BMP, among others), tumor-derived factors that stimulate bone resorption (PTHrP, TGF-β, among others), and tumor-derived factors that affect bone formation (BMP, IGFs, among others), while PTHrP stimulates RANKL production by osteoblasts.

Tumor treatments may affect healthy cells as well. Anti-resorptive therapy targets osteoclasts, when an osteolytic metastatic bone disease is present. Bisphosphonates (e. g., alendronate or zoledronic acid) [50, 9] and monoclonal antibodies (e. g., denosumab) [44, 16] are effective treatments currently employed. Bisphosphonates lodge in bone and poison osteoclasts as they degrade bone. Monoclonal antibodies bind exclusively to RANKL, increasing the OPG/RANKL ratio and inhibiting osteoclast formation. For other diseases, such as multiple myeloma (MM), therapies include daily doses of PTH, endothelin, and proteasome inhibitors, which act by targeting osteoblasts to recover bone mass [28]. Anti-cancer agents that target metastatic and primary tumor cells directly (chemotherapy, as paclitaxel [30], and hormone therapy) should be used in combination with the aforementioned therapies in either case [8].

2.2 Pharmacokinectics and pharmacodynamics PK models characterize drug absorption and disposition [15]. The remaining drug concentration to be absorbed (Cg , in mg/L) and the effective drug concentration in the

74 | J. P. Neto et al. plasma (Cp , in mg/L) are described by D1 Cg (t) = −κg Cg (t),

(1a)

D Cp (t) = κg Cg (t) − κp Cp (t).

(1b)

1

Here κg and κp are the absorption and elimination rate, respectively [23]. For subcutaneous drug administration, the initial concentration is applied in the remaining drug to be absorbed (Cg (0) = C0 ). For the intra-venous case, the initial concentration goes directly to the plasma (Cp (0) = C0 ). For a single drug dosage of initial concentration C0 , plasma concentration can be determined by equation (2a). For multiple doses, the plasma concentration of the nth dose is given by equation (2b), D F for initial conditions C0 = V0 administered at equally spaced intervals t 󸀠 = t −(n−1)τ, d where D0 is the dosage, F the bioavailability, and Vd the volume distribution. Multiple C dosages are governed by the steady state Cpss = τκ0 . We have p

Cp = C0 Cp (n, t 󸀠 ) = C0

κg

κg − κp κg

κg − κp

(e−κp t − eκg t ), ×(

(2a)

1 − e−nκp τ −κp t 󸀠 1 − e−nκg τ −κg t 󸀠 − ). e e 1 − e−κp τ 1 − e−κg τ

(2b)

PD influences a drug’s effect. It can be described by a Hill function that depends Cp (t) on the drug’s concentration, given by d(t) = C50 (t)+C [31]. It varies between 0 and 1, (t) p

p

where Cp50 (t) = f (t)Cp50/base represents the concentration at 50 % of its maximum ef-

fect, Cp50/base is the initial value of Cp50 (t), and resistance to a drug can be described by t

f (t) = 1 + Kr ∫0 max[0, Lr − Cp (λ)]dλ [31]. Different drugs can act on the same pathway. Their combined action may differ from the sum of the effects of the administration of each separate drug. To quantify the synergistic or antagonistic effect of the combination of drugs, the combination index (CI) is used, i. e., CI =

(Cp∗ )1

(C50 )1 d/(1 − d) (C )

dc12 =

[ (C p )1 + 50 1

50 1

(Cp∗ )2

(C50 )2 d/(1 − d)

(Cp )2 ] (C50 )2

CI + [ (C p )1 + (C )

+

(Cp )2 ] (C50 )2

.

,

(3a) (3b)

∗ Here Cpi is the necessary concentration of drug i, when combined with other drugs, to produce the same effect d as concentration Cpi of drug i when taken alone. The resulting combined effect of two drugs is given by dc12 . A PK/PD model combines a PK model and a PD model. Figure 2 shows a simulation of a PK/PD model for multiple doses of a single drug (administered orally or subcutaneously). A drug pathway can have an inhibitory (i: −) or a stimulatory (s: +) effect on a given metabolism. Consequently, a control action (CA) for the tumorous presence in

Variable-order derivatives and bone remodeling in the presence of metastases | 75

Figure 2: Multiple dosage administration. Left: evolution of the PK model; Right: evolution of the corresponding PD case. Used parameters were Cg (0) = 0.008 mg/L, κg = 4 days−1 , κp = 0.5 days−1 , τ = 1 day, Cpss = 0.016 mg/L, and Cp50 (t) = 0.003 mg/L.

the mathematical models for bone remodeling is given by CA(t) = 1 ± Ki,s d(t), where constants Ki , Ks > 0 represent the maximum effect of a drug in a specific mechanism; d(t) is the PD response of a single drug or a combination of drugs.

2.3 Mathematics of bone remodeling – reviewing integer models Different models have been created to analyze and simulate biochemical interactions between the tumor cells with the bone micro-environment. They can be divided in local and non-local constructions (mainly one-dimension models). Within each of these categories, there are three stages of bone behavior: healthy bone micro-environment, tumor-disrupted bone dynamics, and therapy counteraction. Here only models based upon differential equations, used for fractional- and variable-order calculus approaches, are presented, but Table 1 provides a road map into other published models. In [19] the simplest model for bone remodeling was proposed. Taking an S-system form, the coupling of osteoclast, C(t), and osteoblast, B(t), behavior is done through biochemical autocrine (gCC , gBB ) and paracrine (gBC , gCB ) factors expressed implicitly in the system’s exponents. Bone mass density, z(t), is determined through the extent to which values of active C(t) and B(t) populations are above the non-trivial steady state, CSS and BSS . Consequently, bone mass is but the reflection of the bone cell activities. Production and death rate of the bone cells are encompassed in αC,B and βC,B , respectively. Constants κC and κB represent bone resorption and formation activity, respectively. This model is capable of representing single or periodical remodeling cycles by

76 | J. P. Neto et al. Table 1: Published models for biochemical bone remodeling. A healthy bone environment, a bone disrupted by a tumor, and bone with tumor counteracted with therapy are considered. The * indicates models with spatial evolution. Intermediate lines indicate no dependency between papers. Published models [19]

[20]

[18] [51] [17] [12] [2]*

[34] [33]

[12] [11]*

[6]* [32]* [49]

[39]* [38]* [40]* [1]*

Healthy

Tumor

Tumor & treatment

√ √ √ √ √ √ – – √ √ √ √ √ – √ √ – √

– – – – √ √ – – – – – – – √ – – √ √

– – – – √ √ √ √ – – – – – – – – – √

setting the autocrine and paracrine parameters to the appropriate values, specially the osteoblast-derived osteoclast paracrine regulator gBC . The RANK/RANKL/OPG pathway is also implicitly encoded in this parameter. Response amplitude and frequency depend on initial conditions, triggered by a deviation from the steady state. As to disruptive pathologies to the bone micro-environment, [2] extends the model of [19] incorporating the effect of MM in the bone dynamics; T(t) represents the tumor cell density at time t, with a Gompertz form of constant growth γT > 0, and acts through the autocrine and paracrine regulatory pathways in the form of rij parameters. The tumor action is considered independent of the bone mass, with a possible maximum tumor size of LT . Periodic remodeling cycles are deregulated and bone mass density decreases. Tumor treatment was also proposed. In [13], a treatment of osteolytic bone metastases through anti-cancer and anti-resorptive therapy is proposed, adapting the model of [2]. It corresponds to the administration of chemotherapy (d3 ) combined with either bisphosphonates (d2 ) or monoclonal antibodies (d1 ). Bisphosphonates (e. g., zoledronic acid or alendronate) promote osteoclast apoptosis, and monoclonal antibodies (e. g., denosumab) indirectly inhibit osteoclast formation by acting as a decoy receptor for RANKL. Together with chemotherapy (e. g., paclitaxel), the drug effect of this treatment was included in the model through their PK/PD action.

Variable-order derivatives and bone remodeling in the presence of metastases | 77

is

The complete model, which incorporates all three bone micro-environment cases, gCC (1+rCC T(t) ) L

D1 C(t) = αC C(t)

T

gBC (1+rBC T(t) )(1+Ks1 d1 (t)) L

B(t)

T

− (1 + Ks2 d2 (t))βC C(t),

( D1 B(t) = αB C(t) 1

gCB T(t) 1+rCB LT

)

(4a)

(g −r T(t) ) B(t) BB BB LT − βB B(t),

(4b)

D z(t) = −κC max[0, C(t) − Css ] + κB max[0, B(t) − Bss ],

(4c)

D1 T(t) = (1 − Ki3 d3 (t))γT T(t) log (

(4d)

LT ). T(t)

Simulations can be found in Figures 3–4, where parameters are the same as in [2], and treatment values follow published literature; all can be found in Table 5. The previous model can be extended to include diffusion processes in the bone through partial differential equations. The model in [2] is also extended by allowing 𝜕2 the diffusion over one dimension, σi 𝜕x 2 , of osteoclasts, osteoblasts, bone mass, and tumor (MM). They now depend on both t and x ∈ [0, 1]. The diffusion of z accounts for the stochastic nature of bone dynamics and not necessarily migration of cells. All variables are subject to null Neumann boundary conditions. Initial conditions, now depending on both t and x, can be found in [2] (C(0, x) and T(0, x)). In [13], the model

Figure 3: Osteoclasts, osteoblasts, and bone mass evolutions, for the model of equations (4). Full lines represent periodic cycles of remodeling, for a healthy bone (C(0) = 11.76, Css = 1.16, B(0) = Bss = 231.76, T (t) = 0, di (t) = 0, for i = 1, 2, 3). Dashed lines address the plain tumor case, without application of metastasis treatment (di (t) = 0). The black lines, for all three populations, represent the respective stationary state. Used parameters follow the work of [2] and can be found in Table 5. The corresponding tumor evolution is represented, in blue, on the left-side graphic of Figure 7.

78 | J. P. Neto et al.

Figure 4: Osteoclasts, osteoblasts, and bone mass evolutions, for the model of equations (4). Full lines represent the PK/PD combination of chemotherapy (paclitaxel – d3 (t)) with monoclonal antibodies (denosumab – d1 (t)). Dashed lines represent the combination of chemotherapy with bisphosphonates (zoledronic acid – d2 (t)). Again, black lines represent the stationary state. For both cases, treatment begins at tstart = 600 days and it is interrupted at tstop = 2340 days. Used parameters follow the work of [2], and PK/PD treatment parameters follow published literature (both can be found in Table 5). Tumor evolution counteracted with chemotherapy (d3 (t)) is represented by a dashed green line, on the left-side graphic of Figure 7.

of [2] is further extended to include the PK/PD action of anti-cancer and anti-resorptive therapy in the one-dimensional model. The complete non-local model, encompassing all three bone micro-environment possibilities, is 𝜕2 C(t, x) − (1 + Ks2 d2 (t))βC C(t, x) 𝜕x 2 g (1+r T(t) ) g (1+r T(t) )(1+Ks1 d1 (t)) + αC C(t, x) CC CC LT B(t, x) BC BC LT ,

D1 C(t, x) = σC

𝜕2 B(t, x) − βB B(t, x) 𝜕x 2 ( gCBT(t) ) (g −r T(t) ) 1+r + αB C(t, x) CB LT B(t, x) BB BB LT ,

(5a)

D1 B(t, x) = σB

D1 z(t, x) = σz

𝜕 z(t, x) − κC max[0, C(t, x) − Css (x)] 𝜕x 2 + κB max[0, B(t, x) − Bss (x)],

D1 T(t, x) = σT

(5b)

2

2

L 𝜕 T(t, x) + (1 − Ki3 d3 (t))γT T(t, x) log ( T ). 2 T(t, x) 𝜕x

(5c) (5d)

Figure 5 represents both a healthy bone micro-environment (first row) and a tumordisrupted one (second row). Figure 6 represents PK/PD therapy counteraction by com-

Variable-order derivatives and bone remodeling in the presence of metastases | 79

Figure 5: Non-local simulation of osteoclasts, osteoblasts, and bone mass, for equations (5). First row, for healthy remodeling cycles (B(0) = Bss = 231.76, T (t, x) = 0, and di (t) = 0 for i = 1, 2, 3); Second row, for a tumor-disrupted bone micro-environment without applied treatment (di (t) = 0). Parameters and initial and boundary conditions follow exactly what was presented in [2], including the spatial distribution of C(0, x) and T (0, x) (Table 5). Untreated tumor evolution is presented in the second graphic of Figure 7.

bined chemotherapy (d3 (t)) with monoclonal antibodies (first row, d1 (t)) or bisphosphonates (second row, d2 (t)). Parameters are again the same as in [2]; treatment values follow published literature and both can be found in Table 5. Variables and parameters found in equations (4) and (5) are summarized in Tables 2 and 3. Table 2: Summary and description of the variables of the models of equations (4)–(5). Variable

Description

Unit

t x C(t, x) B(t, x) z(t, x) T (t, x) α(t)/α(t, x) d1 (t) d2 (t) d3 (t)

Time Distance Osteoclast population Osteoblast population Bone mass density Bone metastasis density Variable order expression Effect of denosumab Effect of zoledronic acid Effect of paclitaxel

day x ∈ [0, 1] – – % % – – – –

80 | J. P. Neto et al.

Figure 6: Non-local simulation of osteoclasts, osteoblasts, and bone mass, for equations (5) with PK/PD applied therapy. First row, chemotherapy (paclitaxel – d3 (t)) is combined with monoclonal antibodies (denosumab – d1 (t)); Second row, the same chemotherapy is combined with bisphosphonates (zoledronic acid – d2 (t)). For both cases, treatment begins at tstart = 980 days and it is interrupted at tstop = 2340 days. Again, parameters and initial and boundary conditions follow exactly what was presented in [2], and PK/PD treatment parameters follow published literature (both can be found in Table 5). Corresponding tumor evolution is presented in the right-side graphic of Figure 7.

Figure 7: Local (equations (4)) and non-local (equations (5)) tumor evolutions. Left-side graphic: local evolution of the tumor’s density, untreated (dashed green line) and treated with PK/PD chemotherapy of paclitaxel (d3 (t)) (full blue line, tstart = 600, tstop = 2340). Middle graphic: untreated spatial tumor evolution for an initial location on the right side of the normalized bone. Rightside graphic: tumor counteracted with the same chemotherapy action of paclitaxel (tstart = 980, tstop = 2340). Anti-tumor therapy parameters, either local or non-local, follow Table 5. In the case of models with fractional derivatives, units day−1 are replaced by pseudo-units day−α .

Variable-order derivatives and bone remodeling in the presence of metastases | 81 Table 3: Summary and description of the parameters of the models of equations (4)–(5). OC stands for osteoclasts, OB stands for osteoblasts. Parameter

Description

Unit

αC αB βC βB gCC gBC gCB gBB κC κB σC σB σz σT

OC activation rate OB activation rate OC apoptosis rate OB apoptosis rate OC autocrine regulator OC paracrine regulator OB paracrine regulator OB autocrine regulator Bone resorption rate Bone formation rate Diffusion coefficient for OC Diffusion coefficient for OB Diffusion coefficient for bone mass Diffusion coefficient for metastases

day−1 day−1 day−1 day−1 – – – – day−1 day−1 day−1 day−1 day−1 day−1

rCC rBC rCB rBB LT γT

OC tumorous autocrine regulation OC tumorous paracrine regulation OB tumorous paracrine regulation OB tumorous autocrine regulation Maximum size of bone metastases Metastases growth rate

– – – – % % day−1

Css Bss

Steady-state OC number Steady-state OB number

– –

C(0)/C(0, x) B(0)/B(0, x) z(0)/z(0, x) T (0)/T (0, x)

D0 τ F Vd κg κp Cp50/base Kr Ks1 Ks2 Ki3 θ

Initial distribution of osteoclasts Initial distribution of osteoblasts Initial bone mass percentage Initial tumorous mass percentage

Drug dosage Drug administration time interval Bioavailability Volume distribution Drug absorption rate Drug elimination rate Initial drug concentration for 50 % of its maximum effect Drug resistance capacity Maximum effect of denosumab Maximum effect of zoledronic acid Maximum effect of paclitaxel Variable order coefficient

– – % %

mg day – L day−1 day−1 mg/L – – – – –

82 | J. P. Neto et al. More recently, [12] extended the model of [19], by including new variables that represent the pooled concentration of PTH and PTHrP. The former is included in PTHpool (t), where δ(t) is a train of Dirac deltas occurring stochastically with a Weibull distribution with probability P(δ(t) = 1). This initiates a single remodeling cycle, inducing an increase in PTH concentration. The production of RANKL by osteoblasts increases, thus affecting gBC . It also allows for the disruptive action of an osteolytic tumor to be included. Osteoblasts have the same receptors for both PTH and PTHrP, which contribute to the increased production of RANKL. This means the number of osteoclasts will rise, resulting in an increased bone resorption. Bone resorption, in turn, releases tumor growth factors from bone, such as TGF-β, allowing the tumor to grow and to produce PTHrP. The influence of growing bone metastasis, in the bone micro-environment, is included by encoding the production of PTHrP by the metastatic cells, by adding rPTHrP max[0, C(t) − Cth (t)] T(t) to the dynamic expression of PTHpool (t). As such, this LT model is able to describe the vicious cycle of bone metastasis through the action of PTHrP. As for the tumor itself, its growth depends on the factors released by the bone matrix in the resorption phase. Hence, it depends on the osteoclast population (max[C(t) − Cth (t)]) and it has a saturation term introduced by a sigmoid function. Besides affecting the number of osteoclasts indirectly through PTHpool (t), the tumor also affects the dynamics of the system through the autocrine parameters, similarly to [19]. In the presence of a tumor, the thresholds for active osteoclasts and osteoblasts are no longer the static of Css and Bss , since the steady state is affected thereby. Dynamic thresholds Cth (t) and Bth (t) are then used to determine the number of active cells. This model also adapts the previous pharmacological PK/PD treatment structure of anti-cancer (chemotherapy – paclitaxel, d3 (t)) and anti-resorptive therapy (monoclonal anibodies – denosumab, d1 (t); or bisphosphonates – zoledronic acid, d2 (t)). The model is gCC +rCC T(t) L

D1 C(t) = αC C(t)

T

gBC +KPTHpool

B(t)

BC

PTHpool (t)−Ks1 d1 (t)

− (βC + Ks2 d2 (t))C(t),

gBB +rBB T(t) L

1

D1 B(t) = αB C(t)gCB B(t)

T

(6a) − βB B(t),

D PTHpool (t) = −βPTH PTHpool (t) + KPTH δ(t) + rPTHrP max[0, C(t) − Cth (t)] P(δ(t) = 1) = 1 − exp(−(

k

t w ) ), λw

D1 T(t) = κT max[0, C(t) − Cth (t)]

T(t) , LT

(6b) (6c) (6d)

T(t) − Kd3 di3 (t)T(t), λT + T(t)

(6e)

Variable-order derivatives and bone remodeling in the presence of metastases | 83

D1 z(t) = −κC max[0, C(t) − Cth (t)] + κB max[0, B(t) − Bth (t)], gCC +rCC T(t) LT

D1 Cth (t) = αC Cth (t)

Bth (t)gBC −Ks1 d1 (t)

− (βC + Ks2 d2 (t))Cth (t),

gBB +rBB T(t) LT

D1 Bth (t) = αB Cth (t)gCB Bth (t)

(6f) (6g)

− βB Bth (t).

(6h)

Its variables and parameters are summarized in Tables 2, 3, and 4. Simulations for this model are presented in Figure 8 and in Figure 9. Table 4: Summary and description of the variables and parameters of the model of equations (5), as a complement to Tables 2 and 3. OC stands for osteoclasts, OB stands for osteoblasts. In the case of models with fractional derivatives, units day−1 are replaced by pseudo-units day−α . Variable

Description

Unit

PTHpool (t) Cth (t) Bth (t)

PTH/PTHrP concentration variation Dynamic threshold for active OC Dynamic threshold for active OB

ng/L – –

Parameter

Description

Unit

KPTHpool

Influence of PTH/PTHrP on RANKL/OPG ratio

ng−1 L

PTH/PTHrP degradation rate PTH growth rate Rate of PTHrP production by cancer cells Scale parameter of Weibull distribution Shape parameter of Weibull distribution Bone metastases growth rate through bone resorption Half-saturation constant for bone metastases size

day−1 ng−1 L ngL−1 day−1 – – % day−1 %

βPTH KPTH rPTHrP λw kw κT λT

BC

3 Fractional- and variable-order models – creating compact biochemical bone remodeling models 3.1 Models with fractional derivatives Fractional- and variable-order derivatives have been successfully used in modeling the dynamics of bone remodeling. This is reasonable, as anomalous diffusion is often found in biological processes and corresponds to partial differential equations with fractional derivatives in order to time, and second-order derivatives in order to space [24]. Consequently, in [11] fractional derivatives in the differential equations of bone remodeling of [2] are implemented. They analyze the dynamic bone remodeling behavior in the absence and presence of tumor and treatment, for a discretized single

84 | J. P. Neto et al.

Figure 8: Osteoclasts, osteoblasts, PTH/PTHrP concentration, and bone mass evolutions, for the model of equations (6). Each remodeling cycle is triggered by an increase in PTH, for a healthy bone micro-environment (T (t) = 0, di (t) = 0, for i = 1, 2, 3). Used parameters follow the work of [12] and can be found in Table 5.

point, and also for a one-dimensional bone. The latter case is a modification of diffusion model (5), where anomalous diffusion results in the following equations: 𝜕2 C(t, x) − (1 + Ks2 d2 (t))βC C(t, x) 𝜕x 2 g (1+r T(t) ) g (1+r T(t) )(1+Ks1 d1 (t)) + αC C(t, x) CC CC LT B(t, x) BC BC LT ,

Dα C(t, x) = σC

Dα B(t, x) = σB

(7a)

2

𝜕 B(t, x) − βB B(t, x) 𝜕x 2 ( gCBT(t) ) (g −r T(t) ) 1+r + αB C(t, x) CB LT B(t, x) BB BB LT ,

Dα z(t, x) = σz

𝜕 z(t, x) − κC max[0, C(t, x) − Css (x)] 𝜕x 2 + κB max[0, B(t, x) − Bss (x)],

Dα T(t, x) = σT

(7c) b

𝜕2 T(t, x) ) , T(t, x) + (1 − Kd34 d34 (t))γT T(t, x)a ( LT 𝜕x 2

Dα Cg (t) = −ka Cg (t), α

D Cp (t) = ka Cg (t) − ke Cp (t), d(t) =

(7b)

2

Cp (t)

C50 + Cp (t)

.

(7d) (7e) (7f) (7g)

To show more clearly the effect of a different order α, Figure 11 shows simulation results for this model in the case of a healthy bone without dimensions (and with-

Variable-order derivatives and bone remodeling in the presence of metastases | 85

Figure 9: Osteoclasts, osteoblasts, and PTH/PTHrP concentration, for the complete model of equations (6). Full yellow lines represent the PK/PD combination of chemotherapy (paclitaxel – d3 (t)) with monoclonal antibodies (denosumab – d1 (t)). Dashed purple lines represent the combination of chemotherapy with bisphosphonates (zoledronic acid – d2 (t)). For both cases, treatment begins at tstart = 2000 days and it is interrupted at tstop = 3000 days. Used parameters follow the work of [12], and PK/PD treatment parameters follow published literature (Table 5). Corresponding bone mass and tumor evolutions can be found in Figure 10.

Figure 10: Bone mass and tumor evolutions, for the complete model of equations (6), following the results of Figure 9.

86 | J. P. Neto et al. Table 5: All variables and parameters used for simulation of the complete models with an acting tumor. Parameter σi encompasses i = C, B, T , z. PK/PD models, for all models, included monoclonal antibodies, d1 (denosumab), bisphosphonates, d2 (zoledronic acid) and anti-cancer therapy, d3 (paclitaxel). PK parameters (D0 , τ, F , Vd , κg , and κp ) for denosumab can in found in [16, 44], those for zoledronic acid in [50, 9], and those for paclitaxel in [30]. PD parameters, Cp50/base , and Ks,i , were chosen through simulation.

Par.

(4)

(5)

(6)

(7)

αC αB βC βB gCC gBC gCB gBB rCC rBC rCB rBB σi κC κB θ

3 4 0.2 0.02 1.1 −0.5 1.0 0 0.005 0 0 0.2 – 0.0748 6.39 × 10−4 –

3 4 0.2 0.02 1.1 −0.5 1.0 0 0.005 0 0 0.2 10−6 0.45 0.0048 –

3 4 0.2 0.02 0.1 −1 0.4 0.2 0.007 0 0 −0.063 – 0.5 0.0025 –

3 4 0.2 0.02 1.1 −0.5 1.0 0 0.005 0 0 0.2 – 0.1548 6.39 × 10−4 4 × 10−7

PK/PD Par. D0 τ F Vd κg κp Cp50/base Kr Ks,i

Par. γT LT C(0, x) B(0, x) T (0, x) z(0, x) Css Bss βPTH KPTH rPTHrP λw kw κT λT

(4)

(5)

(6)

(7)

0.005 100 15 316 1 100 5 316 – – – – – – –

0.004 100 [2] 316 [2] 100 5 316 – – – – – – –

– 100 0.061 185.8123 1 100 – – [0.1] [1] [0.004] [300] [15] [1] [10]

0.005 100 15 316 1 100 5 316 – – – – – – –

d1

d2

d3

120 28 0.62 3.1508 0.2568 0.0248 1∗,∗∗ , 1.2∗∗∗ – 0.004∗,∗∗ , 0.48∗∗∗

4 28 1 536.3940 – 0.1139 0.0001 – 0.058∗,∗∗ , 1.2∗∗∗

176 7 1 160.2570 – 1.2797 0.0002∗.∗∗ , 0.002∗∗∗ – 1.70

out diffusion). Figure 12 shows how the tumor grows for this fractional-order model, when no treatment is applied. The results of combining chemotherapy with monoclonal antibodies are given in Figure 13, and those of combining chemotherapy with bisphosphonates in Figure 14. Simulations for a one-dimensional bone show the same effect of the order α in the results, as discussed in Section 4. Parameters are given in Table 5. Notice that the equations of the PK model from Section 2.2 were assumed to have the same fractional order as the model itself. Such adaptation is given by equations (7e)–(7f). The PD component (7g), however, remains unaltered.

Variable-order derivatives and bone remodeling in the presence of metastases | 87

Figure 11: Osteoclasts, osteoblasts, and bone mass evolutions, for the model of equations (7), in the absence of tumor. Used parameters follow the work of [11] and can be found in Table 5.

Figure 12: Osteoclasts, osteoblasts, bone mass, and tumor evolutions, for the model of equations (7), in the absence of treatment. Used parameters follow the work of [11] and can be found in Table 5.

88 | J. P. Neto et al.

Figure 13: Osteoclasts, osteoblasts, bone mass, and tumor evolution, for the model of equations (7), with treatment combining chemotherapy with bisphosphonates. Used parameters follow the work of [11] and can be found in Table 5.

Figure 14: Osteoclasts, osteoblasts, bone mass, and tumor evolutions, for the model of equations (7), with treatment combining chemotherapy with monoclonal antibodies. Used parameters follow the work of [11] and can be found in Table 5.

Variable-order derivatives and bone remodeling in the presence of metastases | 89

Also notice that equations (4) and (5), following [2], model the growth of the tumor T with a Gompertz curve, with maximum size LT and a growth rate γT affecting a logarithmic term. This function, however, is not well-defined for T = 0. It is replaced in equation (7) by a logistic function, which has a stable equilibrium point at T = 0 and a stable equilibrium point at T = LT . Model (6) was likewise adapted with the inclusion of anomalous diffusion by [46] as follows: Dα C(t, x) = σC

𝜕2 C(t, x) − (βC + Ks2 d2 (t))C(t, x) 𝜕x 2 gCC +rCC T(t,x) LT

+ αC C(t, x)

gBC +KPTHpool

B(t, x)

(8a) BC

PTHpool (t)−Ks1 d1 (t)

,

𝜕2 g +r T(t,x) B(t, x) + αB C(t, x)gCB B(t, x) BB BB LT − βB B(t, x), 2 𝜕x Dα PTHpool (t, x) = −βPTH PTHpool (t, x) + KPTH δ(t) Dα B(t, x) = σB

+ rPTHrP max[0, C(t, x) − Cth (t, x)] P(δ(t) = 1) = 1 − exp(−( Dα T(t, x) = σT

k

T(t, x) , LT

t w ) ), λw

(8b)

(8c) (8d)

𝜕2 T(t, x) − Kd3 di3 (t)T(t, x) 𝜕x 2

+ κT max[0, C(t, x) − Cth (t, x)]

T(t, x) , λT + T(t, x)

𝜕2 z(t, x) − κC max[0, C(t, x) − Cth (t, x)] 𝜕x 2 + κB max[0, B(t, x) − Bth (t, x)],

(8e)

Dα z(t) = σz

gCC +rCC T(t,x) LT

Dα Cth (t, x) = αC Cth (t, x)

Bth (t, x)gBC −Ks1 d1 (t)

− (βC + Ks2 d2 (t))Cth (t, x),

gBB +rBB T(t,x) LT

Dα Bth (t, x) = αB Cth (t, x)gCB Bth (t, x)

− βB Bth (t, x).

(8f) (8g) (8h)

The qualitative conclusions about the influence of anomalous diffusion and of the fractional order α that can be taken from this model are similar to those of model (7); so, no simulations are shown here.

3.2 Models with variable-order derivatives In [25, 26], and [27], variable-order derivatives have been introduced as a simplification technique in the same models of [2] and [13], in an effort to replicate the same bone micro-environment response but recurring to less parameters to impose the known bone behavior. It turns out that the effects of an osteolytic metastasis, given in all

90 | J. P. Neto et al. models presented thus far by changes in the exponents gCC , gCB , gBC , and gBB , can be more easily modeled changing the order of the differential equation. In this case, the integer-order model corresponds to the healthy bone, and anomalous diffusion is employed for the changes induced by the metastasis. Treatments correspond to a recovery of the original differentiation order. This model is given by 𝜕2 C(t, x) − (1 + Ks2 d2 (t))βC C(t, x) 𝜕x 2 + αC C(t, x)gCC B(t, x)gBC (1+Ks1 d1 (t)) ,

Dα(t,x) C(t, x) = σC

𝜕2 B(t, x) − βB B(t, x) 𝜕x 2 + αB C(t, x)gCB B(t, x)gBB ,

(9a)

Dα(t,x) B(t, x) = σB

Dα(t,x) z(t, x) = σz

(9b)

2

𝜕 z(t, x) − κC max[0, C(t, x) − CSS (x)] 𝜕x 2 + κB max[0, B(t, x) − BSS (x)],

Dα(t,x) T(t, x) = σT

(9c)

2

𝜕 T(t, x) 𝜕x 2

+ (1 + Ki3 d3 (t))γT T(t, x) log ( α(t, x) = 1 − θ × t × T(t, x).

LT ), T(t, x)

(9d) (9e)

Figure 15 shows simulation results for the two usual cases: one with chemotherapy and anti-resorptive therapy given by monoclonal antibodies; another with chemother-

Figure 15: Non-local simulation of osteoclasts, osteoblasts, and bone mass, according to the variable-order model of equations (9). Top: PK/PD treatment of anti-cancer (chemotherapy – paclitaxel d3 ) and anti-resorptive therapy (monoclonal antibodies – denosumab d1 ); Bottom: PK/PD treatment of anti-cancer (chemotherapy – paclitaxel d3 ) and anti-resorptive therapy (bisphophonates – zoledronic acid d2 ). Parameters and initial and boundary conditions follow [2] and are found in Table 5. The corresponding tumor evolution is presented in Figure 16 and is identical for both therapies.

Variable-order derivatives and bone remodeling in the presence of metastases | 91

Figure 16: Non-local tumor evolution, counteracted with the PK/PD chemotherapy action of paclitaxel (equation (9d)).

apy and anti-resorptive therapy given by bisphosphonates. Parameters are given in Table 5. A straightforward extension of this model for three dimensions is presented in [48].

4 Discussion Fractional derivatives incorporate anomalous diffusion in bone remodeling models and, moreover, make them more versatile. The qualitative behavior of fractional- and variable-order models is similar to that of integer-order models, but it is possible to better shape the period and amplitude of the oscillations and the effects of the tumor and of recovery with treatment. As might be expected, larger values of differentiation order α correspond to faster and more oscillatory systems; lower values of α, to more stable and sluggish behaviors. Furthermore, variable-order models describe the same behavior but with less parameters, and in a manner easier to interpret. It should be stressed that parameter tuning for these models has until now been always done based upon a mostly qualitative assessment of the results. Experimental data to find values for the coefficients of the models may in the future be extrapolated from experiments with animals (human data are likely too expensive, and, more importantly, unethical, due to the invasiveness of collection procedures), or result from medical imaging techniques [4]. This is probably the biggest challenge facing the mathematical study of bone remodeling. Different parameters can in the future be available for particular patients, allowing treatments adjusted for each case (personalized medicine). The analysis and simulation of these models may thus provide a better planning of cancer treatments, for the relief of the patients diagnosed each year.

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P. Melchior, S. Victor, M. Pellet, J. Petit, J.-M. Cabelguen, and A. Oustaloup

Skeletal muscle modeling by fractional multi-models: analysis of length effect

Abstract: Many experiments and measures have been carried out on skeletal muscles of different species, such as frog, salamander, rabbit, and rat. Exploiting these measures for system identification through non-integer models has led to many promising results. These models are useful in medicine as they help understanding their organ working or dysfunction. Moreover, they are used for the rehabilitation of deficient muscles via electrical stimulation. Finally, these models are particularly interesting for bio-inspired robotics; indeed, they enable piloting an actuator in order to more precisely reproduce a natural movement of an articulation or a limb. Non-integer modeling has the advantage of ensuring parametric parsimony (or parametric compactness), thus enabling an easier implementation (“embeddable” model). A muscle is composed of muscular fiber groups that are innervated by several motor neurons called α motor neurons. The motor units are classified as three different kinds according to their physiological and contractile properties: motor units of type S (Slow), motor units of type FF (Fast, Fatigable), and motor units of type FR (Fast, Resistant). A non-integer multi-model is able to predict the time response of a motor unit (whatever its type) to an electrical stimulation during an isometric contraction. In order to make the multi-model more representative of the real dynamics of a motor unit, the muscle length influence on the model parameters has been studied to determine the best strategy to take into consideration this phenomenon. Keywords: fractional differentiation, fractional system identification, biological systems, multi-models, skeletal muscle modeling PACS: 93B30,93E24,26A33,34A08,78A70,34K37

P. Melchior, S. Victor, M. Pellet, A. Oustaloup, Univ. Bordeaux, CNRS, IMS, UMR 5218, Bordeaux INP/enseirb-matmeca, 351 Cours de la Libération, Bat. A31, F33405 Talence cedex, France, e-mails: [email protected], [email protected], [email protected], [email protected] J. Petit, Univ. Bordeaux, CNRS, Laboratoire Mouvement Adaptation Cognition, UMR 5227, Zone Carreire Nord Bat. 2A, 146 Rue Léo Saignat, 33076 Bordeaux cedex, France, e-mail: [email protected] J.-M. Cabelguen, INSERM E 0358, Institut Magendie, A Rue Camille St Saëns, 33077 Bordeaux cedex, France, e-mail: [email protected] https://doi.org/10.1515/9783110571905-006

96 | P. Melchior et al.

1 Introduction Fractional calculus is widely used in bio-engineering [14, 29], and more specifically in lung modeling [8–10], human arm modeling [2, 26], and muscle modeling [11, 12]. Exploiting non-integer differentiation as a modeling tool of skeletal muscles has been initiated by the CRONE team through two previous PhD theses [13, 22]. Many experiments and measures have been carried out on skeletal muscles of different species, such as frog, salamander, rabbit, and rat. Exploiting these measures for system identification through non-integer models has led to many promising results. Many more applications are possible for skeletal muscle models. These models are useful in medicine as they help understanding their organ working or dysfunction. Moreover, they are used for the rehabilitation of deficient muscles via electrical stimulation [3, 6, 15, 21]. Finally, these models are particularly interesting for bio-inspired robotics; indeed, they enable piloting an actuator in order to more precisely reproduce a natural movement of an articulation or a limb. In this perspective, authors in [7, 25], in collaboration with the Magendie institute of INSERM and the laboratory MAC of University of Bordeaux 2, have identified a non-integer multi-model of a muscle in order to implant it in a simulator of a bio-inspired salamander robot. This non-integer model has the advantage of ensuring parametric parsimony (or parametric compactness), thus enabling an easier implementation (“embeddable” model). A muscle is composed of muscular fiber groups that are innervated by several motor neurons called α motor neurons. The motor units are classified as three different kinds, according to their physiological and contractile properties: – motor units of type S (Slow) are the slowest ones and produce the weakest contraction force; – motor units of type FF (Fast, Fatigable) are the fastest ones but the most sensitive to fatigue (or tiredness); – motor units of type FR (Fast, Resistant) also are faster than the S types and are less sensitive to tiredness than FF-type motor units. A non-integer multi-model is able to predict the time response of a motor unit (whatever its type) to an electrical stimulation during an isometric contraction. The isometric contraction is a classic concept in muscular physiology. It consists in studying the force evolution produced by a muscle when its length is maintained constant. In this study, only the isometric condition has been considered; isotonic (when the tension in the muscle remains constant despite a change in muscle length) or auxotonic (same as the isotonic contractions with fluctuations towards the end) contractions shall be studied in future works. In order to make the multi-model more representative of the real dynamics of a motor unit, its validity domain should be widened by notably considering several length cases. Indeed, the tension exerted during the muscle activation depends on

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 97

its length [30]. The muscle length influence on the model parameters has therefore been studied in order to determine the best strategy to take into consideration this phenomenon. This chapter is organized as follows: after a brief review of skeletal muscles and the motor units, the experimental protocol will be presented in Section 2. Then the non-integer multi-model of motor units will be detailed in Section 3, to finally present the obtained results on the muscle length influence in Section 4.

2 Motor unit and measures 2.1 Skeletal muscle and motor units The muscle is a tissue characterized by its contractile property. In humans, there exist three types of muscle: the skeletal muscle, the smooth muscle, and the cardiac muscle, each of them exerting a different function in the organism. Skeletal muscles enable the motor function of several parts of the skeleton, which is the reason why they are studied here; the proposed multi-model is intended to simulate systems such as bioinspired robot articulations. Skeletal muscles are organized bundles of muscular fibers, each of the fibers being connected to a neuron, called an α motor neuron, the cell body of which is located inside the spinal cord. The motor neuron innervates several fibers uniformly spread within the same muscle. The set formed by a motor neuron and all the fibers it innervates is called a motor unit. The muscular fibers are classified into three different kinds according to their dynamic and physiological characteristics. Experimentally, the measure of the time contraction is most often used to differentiate the different fiber kinds. The contraction time is defined as the duration between the electrical stimulation of the muscular nerve and the moment the tension developed by the motor unit reaches its maximum. Each motor unit is only composed of a single fiber type. The motor units can then be classified according to the fiber types they are composed of. Therefore there exist three types of motor unit: – motor units of type S (Slow) have the longest contraction time, they produce the lowest tension, and they quickly tire; the organism uses them to make movements of low amplitude with great precision; – motor units of type FF (Fast, Fatigable) have the shortest contraction time but quickly tire; the organism call upon them for reflex movements or short efforts; – motor units of type FR (Fast, Resistant) are also faster than the type S and are less sensitive to tiredness than type FF; therefore the organism calls upon these for sustained efforts.

98 | P. Melchior et al. When an α motor neuron receives an electrical stimulation, it transmits an excitation (called nervous action potential) to all motor unit fibers, without possible selection. Thus, a motor unit forms an elementary contractile fraction of the muscle it belongs to. The electrical stimulations emitted by the motor neurons are of the pulse train kind. The amplitude of the tension generated by the motor unit only depends on its pulse frequency.

2.2 Experimental protocol In physiology, two kinds of experiments are commonly achieved to study the skeletal muscle working (Figure 1). In both cases, the aim is to record the motor unit response to an electrical stimulation of the experimenter. The response of the motor unit can be: – a length variation, recorded while applying a constant force on the muscle; these contractions are called isotonic contractions; – a force, recorded while maintaining a constant length of the muscle; these contractions are called isometric contractions.

Figure 1: Experiment diagram of (a) an isotonic contraction and (b) an isometric contraction.

In this work, only experiments with isometric contractions have been conducted. Our aim was to record the force produced by a motor unit upon different electrical stimulations. In its “natural” working, the electrical stimulations sent by the organism to the muscle are of the pulse train; their frequency range is between 10 and 60 Hz. It is therefore the same kind of pulse train that was used for exciting the motor units.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 99

The studied motor units are those of the peroneus digiti quarti (PDQ) muscle of a Wistar rat. This muscle has been chosen on account of the low number of motor units that it is composed of, thus enabling an exhaustive recording of the muscle elementary units. The animal was anesthetized with a first intra-peritoneal injection of sodium pentoarbital (45 mg/kg), complemented if necessary by other injections to maintain a deep anesthesia. The muscle nerves other than the one studied were severed. A laminectomy was carried out between vertebrae L4 and S2. The skin of this area and of the limb was reflected in a manner to hold trays that were filled with paraffin oil in order to avoid any desiccation. The oil temperature was kept at 37 °C. The dorsal and ventral roots were cut from vertebrae L5 to S1 near their medullar input. They were then divided in thin filaments, and each filament was attached to a silver electrode used as anode. A second electrode used as cathode was in contact with the muscular mass near the root input. The action potentials evoked by the ventral root filament stimulation were detected by electrodes placed at the nerve contact (electromyography, EMG) and amplified by AC Grass amplifiers. The muscle distal tendon was fixed to a force sensor (Kulite BG300, strain gauge type). This same force sensor was assembled on a mobile axis of a translation electrical motor whose position was controlled in order to also control the muscle length. Figures 2 and 3 highlight the two surgical fields of the experiment. The first one was at the animal back level, where the stimulation electrodes were placed in contact of the ventral roots of the spinal cord. The second one was located at the right rear leg, at the studied muscle level. Two measurement series have been carried out to study isometric responses of the three motor unit types: first for different fixed lengths, and then for a varying length.

1 2 3 4

Binocular loop Stimulation electrodes Axis supporting the force sensor controlled in longitudinal position Force sensor

Figure 2: Overall view of the experimental device.

100 | P. Melchior et al.

1 2 3 4 5 6

Stimulation electrode Spinal cord Ventral roots Control electrode recording the EMG Force sensor Studied muscle (PDQ)

Figure 3: Animal back and leg view.

In the first measurement series, the muscle was shortened by 2 mm (length L1 ) from its optimal length (L5 ). The isometric contractions of the motor unit were measured by using two kinds of pulse trains: pulse trains of fixed frequency (5, 10, 20, 40, and 60 Hz) and pulse trains of variable frequency (a triangular frequency modulation as random frequency signal). The muscle was then stretched by 0.5 mm (L2 ) and the isometric contraction force was measured anew. This operation was repeated (L3 = L5 − 1 mm, L4 = L5 − 0.5 mm) until reaching again length L5 . The force signal was sampled at a frequency of 2 kHz and recorded with a 1401plus CED interface coupled to Spike2 software. Experiments have been conducted with three different kinds of motor unit. During the second measurement series, while two pulse trains of fixed frequencies of 10 and 20 Hz were sent to a motor unit, the muscle length was shortened according to the sinus negative half period. System identification results of these measures are presented in Section 4.

3 Motor unit multi-model Thanks to their parametric compactness, non-integer models are particularly well suited to model diffusive systems [18, 19] such as thermal systems [1, 4, 27, 28], biological systems [16, 17, 20, 23, 24], and electrochemical systems [5]. The motor unit model used here has been developed by Sommacal [22]. This model is based on the decomposition of motor unit dynamics into two phases: a con-

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 101

traction phase and a relaxation phase, where each phase is modeled by a submodel M1 (s) or M2 (s).

3.1 Motor unit properties The experimental study established in [22] has revealed two remarkable observations on motor unit time responses. Observation 1. The activation dynamics of the motor unit depends on the past activations. More precisely, the study of the motor unit time responses to pulse train stimulations shows that the duration and maximal amplitude of the contraction phase depend on the reached amplitude at the instant when the muscle receives the pulse. Observation 2. Moreover, for some motor units, if the response to the first pulse is time-variant, different types of prolonged excitations bring, most of the time, the muscle into a final state whose dynamics are unique but whose amplitude is not necessarily unique. These two observations lead to two fundamental hypotheses of multi-models. Hypothesis 1. The duration, gain, and frequency of the contraction phase are functions of the amplitude reached at the instant when the muscle received a pulse. These functions can be approximated by a linear equation. Hypothesis 2. Conditioning the muscle by a pulse train stimulation leads to a repeatable impulse response. This impulse response is chosen as reference for multi-model system identification. In this study, the signal used for the muscle conditioning is the stimulation of triangular frequency modulation type (pulse train whose time period varies between 17 and 200 ms); the last impulse response of this stimulation is chosen to be the reference response.

3.2 Modeling of the motor unit response The multi-model relies on dissociating the contraction and relaxation phases. Indeed, contraction and relaxation depend on different biochemical and mechanical phenomena (release of Ca2+ ions versus pumping of Ca2+ ions), thus opening the possibility to model them separately. The multi-model therefore integrates phase submodels, which enables it to better describe the mechanisms under study. The junction between the two models is of all-or-none type and is carried out after a contraction–relaxation junction time, denoted Tj , corresponding to the time necessary to reach the maximal amplitude of the impulse response. This all-or-none solution takes the place of a hyperbolic tangent type that would highly complicate the multi-model without guaranteeing a lower modeling error.

102 | P. Melchior et al. The submodel associated to the contraction phase is denoted as ℳ1 , and the one associated to the relaxation as ℳ2 . When an impulse is applied, model ℳ1 applies. If no other impulse is applied before the time Tj , the model commutes on ℳ2 and its response then relaxes from the amplitude reached at time t = Tj . When applying another impulse, the submodel ℳ1 is again selected and is summed at the contraction level at which the muscle is when applying this impulse. The response of each submodel is taken into account only when the submodel is active. The principle of the multi-model is illustrated in Figure 4.

Figure 4: Fractional multi-model principle (the contraction phase Tj , the contraction maximal amplitude δA resulting from the ith pulse, and the submodel parameters depend on the amplitude reached at the beginning of each new phase).

3.2.1 Contraction phase modeling The submodel of the phase contraction is identified by using a Davidson–Cole function such that F(s) =

δA

( ωs + 1)

νc

,

(1)

c

where s denotes the Laplace variable and νc ∈ ℝ. In order to take into account Hypothesis 1, parameters δA, ωc , and Tj vary in function of the amplitude Ai reached by the motor unit force when applying the new impulse. The variations of these parameters can be approximated by straight-line equations.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 103

The expression of the submodel ℳ1 can then be written as ℳ1 (s, Ai ) =

δA(Ai ) (ω

s

c (Ai )

+ 1)

νc

,

(2)

with δA(Ai ) = a1 Ai + b1 ,

ωc (Ai ) = a2 Ai + b2 ,

(3)

Tj (Ai ) = a3 Ai + b3 .

The ai and bi coefficients are identified with least squares. Figure 5 illustrates the law variations of the contraction submodel parameters identified for a motor unit of type FR at length L3 .

Figure 5: Variation laws of the contraction submodel, the red dots being the measures and the blue lines the identified variation laws.

3.2.2 Relaxation phase modeling The experimental study of the different relaxation phases has shown that, contrary to the contraction phase, the relaxation dynamics does not depend on the amplitude reached by the force when applying the impulse but on the amplitude from which the force is relaxed. Moreover, as shown in Figures 6 and 7, the relaxation dynamics

104 | P. Melchior et al.

Figure 6: Motor unit responses to four types of input: (a) 10, 20, 40, and 60 Hz; the response to the last impulse of each stimulus is given by part (b); the relaxation phase response (normalized with respect to the initial relaxation) from the last impulse of each stimulus is given by part (c).

Figure 7: Motor unit responses to four types of input: (a) 10, 20, 40, and 60 Hz; the response to the last impulse of each stimulus is given by part (b); the relaxation phase response (normalized with respect to the initial relaxation) from the last impulse of each stimulus is given by part (c).

and form significantly vary when the motor unit relaxes from a low or from a high amplitude. It is therefore difficult to identify the relaxation of a motor unit by a single model. To overcome this problem, the different relaxation phases are modeled as the relaxation phase sum of the reference impulse response (Hypothesis 2) and the difference between the modeled and reference relaxation phase. The block diagram of submodel ℳ2 is given in Figure 8.

Figure 8: Block diagram of submodel ℳ2 .

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 105

In this diagram, model ℳ3 is the relaxation model of the reference impulse response and Δ(s) represents the difference between the modeled and reference relaxation phase of the impulse response. Inputs U1 and U2 are virtual inputs, generated for the system identification and without physical or physiological sense. The expression of these two transfer functions needs to be determined. The relaxation model of the impulse response is expressed by the complementary of a Davidson–Cole function. We have ℳ3 (s) = 1 −

( ωs r

1

+ 1)

νr

(4)

,

the input U1 applied to this transfer function being a step. As for the transfer function Δ(s), the experimental observation has shown that the dynamics varies not much whatever the considered relaxations and can be modeled by the same Davidson–Cole function as the relaxation of the impulse response. Thus Δ(s) = 1 −

( ωs r

1

+ 1)

νr

(5)

,

the input U2 applied to this transfer function being a rectangular function whose amplitude κ and duration τr are functions of the amplitude reached by the muscle force at the instant when the impulse starts the contraction phase, i. e., U2 (s, Ai ) =

κ(Ai ) [1 − e−τr (Ai )s ]. s

(6)

From equations (5) and (6), the transfer function Δ(s) excited by a crenel can know be rewritten into Δ󸀠 (s, Ai ) = κ(Ai )[1 − e−τr (Ai )s ]

( ωs r

1

+ 1)

νr

,

(7)

which is now submitted to the same step input U(s) = U1 (s). The block diagram of submodel ℳ2 (s) can now be represented as in Figure 9.

Figure 9: Simplified block diagram of submodel ℳ2 .

Furthermore, law variations of parameters κ and τr are approximated by straight line equations such as κ(Ai ) = a4 Ai + b4 ,

τr (Ai ) = a5 Ai + b5 .

(8)

106 | P. Melchior et al.

Figure 10: Simplified block diagram of submodel ℳ2 .

Figure 10 illustrates the variation laws of the relaxation submodel parameters identified for a motor unit of type FR at length L3 .

3.2.3 System identification algorithm The block diagram of the multi-model in the case of an impulse response is given in Figure 11.

Figure 11: Principle diagram of the multi-model in the case of an impulse response.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 107

In Figure 11, models ℳ1 and ℳ2 are those established in Section 3.2.1 and Section 3.2.2. The commutation block is such that when an impulse is applied at the input, the submodel ℳ1 stays active as long as no new impulse is applied or as t < Tj . The submodel ℳ2 is activated if no new impulse is applied and t > Tj . The inputs of submodels ℳ1 and ℳ2 are integrated in order to transform the impulses into steps. The zero-order holders, synchronized to the detection of an impulse, enable us to obtain the output value at the beginning of each contraction/relaxation phase in order to initialize the value reached at the preceding phase. The system identification algorithm of the multi-model is composed of four steps: Step 1: System identification of variation laws of parameter δA, ωc , and Tj . Thanks to the data set, δA, ωc , and Tj expressed as in relations (3) are estimated using a linear regression using the least squares method. Step 2: System identification of the contraction phase of the impulse response. Using the last impulse response to a stimulation of triangular frequency modulation type, the values of νc and the y-intercept of equations (3) are estimated to minimize an output error-type criterion. Step 3: System identification of the difference between a high amplitude relaxation and a reference impulse response relaxation. The parameters ωr and νr of submodel Δ󸀠 (s) given in relation (7), and therefore coefficients b4 and b5 of equations (8), are optimized to minimize an output errortype criterion. The used signal is the normalized difference between the last relaxation of the response to a stimulation with a frequency of 60 Hz and the reference relaxation. Step 4: System identification of the variation laws of parameters κ and τr . The two extreme values of the variation laws are computed on a relaxation from a high amplitude (last relaxation of the response to a stimulation with a frequency of 60 Hz, corresponding to maximal values of κ and τr ) and the reference relaxation (last relaxation of the response to a stimulation of triangular frequency modulation type, corresponding to minimal values of κ and τr ).

4 Length influence study As described in Section 2.2, the first campaign consists in measuring the isometric responses of the three motor unit types (FR, FF, and S types) for different muscle lengths. For each muscle length L1 to L4 , the multi-model parameters are estimated by using measurements obtained from a variable-frequency pulse train (triangular frequency modulation) and a pulse train of a fixed frequency of 40 Hz. The input signals are plotted in Figure 12.

108 | P. Melchior et al.

Figure 12: Input signal used for system identification. (a) Triangular frequency modulation pulse train; (b) Pulse train with a frequency of 40 Hz.

4.1 Motor unit estimation results The measured signals for the other stimulation types are used for system identification validation. Four multi-models ML1 , ML2 , ML3 , and ML4 are obtained for each motor unit type, Li , i = 1, . . . , 4, being the muscle length. The responses of the multi-models and their motor units are plotted in Figures 13, 14, and 15. In each figure, the motor unit and its estimated multi-model responses are plotted, from left to right, according to a random-frequency pulse train, a triangular frequency modulation pulse train, a 10-Hz fixed-frequency pulse train, a 20-Hz fixed-frequency pulse train, a 40-Hz fixed-frequency pulse train, and finally a 60-Hz fixed-frequency pulse train. Recalling that the multi-models are given by relations (2), (4), and (7) using the parameters given by relations (3) and (8), the parameter values of each motor unit type (FR, FF, and S) are respectively presented in Tables 1, 2, and 3. The non-integer multi-model is a behavior model in the sense that it should reproduce the motor unit dynamics. In Figures 13, 14, and 15, the modeling error is only given for information purpose, but does not represent a good criterion of the model quality. The true goal of the model, with its thirteen parameters, is to be able to imitate the motor unit responses to six different types of stimulations, and this for the three types of motor units (FR, FF, and S). This new measurement series has shown that the multi-model is also able to reproduce the motor unit dynamics for four different lengths.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 109

Figure 13: Responses of the motor unit (—) and multi-model (—) of type FR with length (a) L1 , (b) L2 , (c) L3 , and (d) L4 .

Table 1: Multi-model parameters for FR-type motor units with different muscle length (Ai being the amplitude reached by the motor unit when applying a new impulse).

δA

ωc

νc

ωr

νr

M L1

M L2

M L3

M L4

−0.35Ai + 0.39

−0.34Ai + 0.37

−0.32Ai + 0.36

−0.28Ai + 0.35

1.43

1.66

1.34

1.76

0.41Ai + 278 146

2.45

0.41Ai + 368 134

2.02

κ

−0.57Ai + 0.71

−0.49Ai + 0.62

Tj

−14.8Ai + 32.0

−13.9Ai + 30.0

τr

17.1Ai + 19.9

24.1Ai + 15.8

0.37Ai + 321 95.8

1.67

−0.92Ai + 1.04

−51.0Ai + 87.6

−12.4Ai + 28.0

0.34Ai + 377 106

1.83

−0.87Ai + 1.03

−45.5Ai + 81.8

−11.5Ai + 26.0

The presented models have been tested using each different muscle length, however static length variations can be taken into account by identifying a single multi-model. Indeed, it is possible to identify parameter variation laws by adding a parameter func-

110 | P. Melchior et al.

Figure 14: Responses of the motor unit (—) and multi-model (—) of type FF with length (a) L1 , (b) L2 , (c) L3 , and (d) L4 .

Table 2: Multi-model parameters for FF -type motor units with different muscle length (Ai being the amplitude reached by the motor unit when applying a new impulse).

δA ωc

νc

ωr

νr

M L1

M L2

M L3

M L4

−0.50Ai + 1.99

−0.49Ai + 2.03

−0.48Ai + 2.03

−0.52Ai + 1.83

1.15

1.21

1.31

1.88

0.09Ai + 251 201

2.68

0.10Ai + 252 199

3.35

0.10Ai + 267 176

2.94

0.10Ai + 394 235

3.49

κ

−0.11Ai + 0.54

−0.12Ai + 0.63

−0.12Ai + 0.62

−0.17Ai + 0.874

Tj

−3.45Ai + 34.0

−3.50Ai + 32.0

−3.37Ai + 32.0

−3.30Ai + 30.0

τr

0.77Ai + 46.4

−0.66Ai + 46.0

0.87Ai + 43.1

0.54Ai + 33.8

tion of the muscle length. Instead of identifying straight-line equations as variation law, plane equations are identified.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 111

Figure 15: Responses of the motor unit (—) and multi-model (—) of type S with length (a) L1 , (b) L2 , (c) L3 , and (d) L4 .

Table 3: Multi-model parameters for S-type motor units with different muscle length (Ai being the amplitude reached by the motor unit when applying a new impulse).

δA

ωc

νc

ωr

νr

M L1

M L2

M L3

M L4

−0.42Ai + 0.89

−0.42Ai + 0.92

−0.42Ai + 0.94

−0.41Ai + 0.91

1.57

1.82

2.11

2.12

0.21Ai + 311 216

3.47

0.18Ai + 352 171

2.81

0.21Ai + 387 171

3.17

0.19Ai + 410 155

2.84

κ

−0.22Ai + 0.62

−0.26Ai + 0.65

−0.14Ai + 0.74

−0.23Ai + 0.80

Tj

−8.52Ai + 33.0

−7.18Ai + 34.0

−8.25Ai + 33.0

−7.06Ai + 31.0

τr

0.11Ai + 45.8

6.32Ai + 46.4

4.10Ai + 44.3

6.50Ai + 36.3

112 | P. Melchior et al.

4.2 Contraction submodel increasing For the contraction submodel, the varying parameters Tj , δA, and ωc are functions of the muscle length L as well as the force amplitude Ai reached when applying the stimulation impulse. Their variation laws are presented in Figure 16.

Figure 16: Variation laws of the parameters Tj , δA, and ωc as functions of the amplitude Ai and the muscle length L for the motor unit of type FR.

The non-integer order νc of the contraction submodel only depends on the muscle length but its variations can hardly be modeled by a straight-line equation; its variation law is estimated by a fourth-order polynomial as plotted in Figure 17. The contraction submodel M1 (s, Ai , L) is written as M1 (s, Ai ) =

δA(Ai , L) , s +1 ω (A ,L) c

(9)

i

Figure 17: Variation law of νc versus the muscle length L.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 113

with δA(Ai , L) = a1 Ai + b1 L + c1 ,

ωc (Ai , L) = a2 Ai + b2 L + c2 ,

(10)

Tj (Ai , L) = a3 Ai + b3 L + c3 ,

νc (L) = a4 L4 + b4 L3 + c4 L2 + d4 L + e4 .

All these parameters are estimated using the least squares method.

4.3 Relaxation submodel increasing In the same manner, the variation laws of the relaxation submodel parameters are revised in order to take into account the muscle length. The variation laws of parameters κ and τr that vary according to the amplitude Ai as well as the length L are plane equations, as shown in Figure 18.

Figure 18: Variation laws of the parameters κ and τr as functions of the amplitude Ai and the muscle length L for the motor unit of type FR.

The variation laws of parameters ωr and νr only depend on the muscle length L but their variations can hardly be modeled by straight-line equations; therefore they are modeled by an interpolation polynomial of degree three, as plotted in Figure 19. The relaxation submodel equations then become M3 (s, L) = 1 −

( ω s(L) r

1 + 1)

νr (L)

,

κ(Ai , L) [1 − e−τr (Ai ,L)s ] ωc (Ai , L) = s (

s ωr (L)

1 + 1)

(11) νr (L)

,

114 | P. Melchior et al.

Figure 19: Variation laws of parameters ωr and νr versus the muscle length L.

with κ(Ai , L) = a5 Ai + b5 L + c5 Ai L + d4 ,

τr (Ai , L) = a6 Ai + b6 L + c6 Ai L + d6 (Ai L)2 + e6 , ωr (L) = a7 L3 + b7 L2 + c7 L + d7 ,

(12)

νr (L) = a8 L3 + b8 L2 + c8 L + d8 .

All these parameters are estimated using the least squares method.

4.4 Augmented multi-model Finally, 18 parameters have been added to the multi-model in order to directly take into account the length variations. The parameter values are presented in Table 4. Table 4: Parameters of the augmented multi-model.

δA ωc νc ωr νr κ τr Tj

ai

bi

ci

di

ei

−0.29 0.37 0.12 4.14 −0.030 −0.43 16.0 −13.0

−0.004 0.016 −1.44 −27.2 0.063 0.12 9.50 −0.84

0.35 1.07 5.85 33.2 0.47 −0.11 −7.70 32.7

− − −9.60 138 −9.60 −0.55 −0.41 −

− − −3.61 − 3.61 − − −

Thanks to its 31 parameters, the multi-model can now directly take into account the muscle length in the simulation of the muscle isometric responses. The parameter number has been highly increased compared with the 13 parameters of the original multi-model valid for only one length, but this number of parameters is nevertheless lower than the total number of parameters when identifying a multi-model for each separate length.

Skeletal muscle modeling by fractional multi-models: analysis of length effect | 115

In the case of isometric contraction experiments, only four lengths have been studied given a motor unit. It is difficult to achieve measures for a higher number of lengths by conserving the same diversity of stimulation. Indeed, repeated experiments on more muscle lengths would tire the muscle, distorting the motor unit dynamics.

5 Conclusion To increase the validity domain of motor unit non-integer multi-models, measurement series have been carried out in order to study the effect of the muscle length on motor unit dynamics. First of all, measurements have been performed under isometric conditions for different muscle lengths, the length being maintained constant during each experiment. These measurements have been used to develop different multi-models, showing that the structure and identification algorithm stay valid whatever the considered working points. Multi-models have been identified for three types of motor units: the FR (Fast, Resistant), FF (Fast, Fatigable), and S (Slow) types. Recalling the experimental protocol, the modeling of the motor unit response is composed of two phases: the contraction phase and the relaxation phase. Finally, during the study of the influence of the length, several multi-models have been identified for each muscle length and for each of the three types of motor unit. The study has been carried out further to propose a single augmented multi-model where the parameters are directly expressed as functions of the pulse amplitude and the muscle length. For further studies, non-integer multi-models can be studied during dynamic length variations.

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Hossein Jafari, Behrouz Mehdinejadiani, and Dumitru Baleanu

Fractional calculus for modeling unconfined groundwater Abstract: The porous medium which groundwater flows in is heterogeneous at all scales. This complicates the simulation of groundwater flow. Fractional derivatives, because of their non-locality property, can reduce the scale effects on the parameters and, consequently, better simulate the hydrogeological processes. In this chapter a fractional governing partial differential equation on unconfined groundwater (fractional Boussinesq equation [FBE]) is derived using the fractional mass conservation law. The FBE is a generalization of the Boussinesq equation (BE) that can be used in both homogeneous and heterogeneous unconfined aquifers. Compared to the BE, the FBE includes an additional parameter which represents the heterogeneity degree of the porous medium. This parameter changes within the range of 0 to 1 in the non-linear form of the FBE. The smaller the value of the heterogeneity degree, the more heterogeneous the aquifer is, and vice versa. To investigate the applicability of the FBE to real problems in groundwater flow, a fractional Glover–Dumm equation (FGDE) was obtained using an analytical solution of the linear form of the FBE for onedimensional unsteady flow towards parallel subsurface drains. The FGDE was fitted to water table profiles observed at laboratory and field scales, and its performance was compared to that of the Glover–Dumm equation (GDE). The parameters of the FGDE and the GDE were estimated using the inverse problem method. The results indicate that one can recognize the heterogeneity degree of porous media examined according to the obtained values for the indicator of the heterogeneity degree. The FGDE and the GDE showed similar performances in homogeneous soil, while the performance of the FGDE was significantly better than that of the GDE in heterogeneous soil. In summary, the FBE can be used as a highly general differential equation governing groundwater flow in unconfined aquifers. Keywords: Fractional Boussinesq equation, fractional Glover–Dumm equation, heterogeneous soil, indicator of the heterogeneity degree, subsurface drain PACS: 47.10.A, 02.30.Jr Hossein Jafari, Department of Mathematical Sciences, University of South Africa, UNISA0003, Pretoria, South Africa; and Department of Mathematics, University of Mazandaran, Babolsar, Iran, e-mail: [email protected] Behrouz Mehdinejadiani, Department of Water Science and Engineering, Faculty of Agriculture, University of Kurdistan, Sanandaj, Iran, e-mail: [email protected] Dumitru Baleanu, Department of Mathematics, Faculty of Art and Sciences, Cankaya University, Ankara, Turkey; and Institute of Space Sciences, Magurele-Bucharest, Romania, e-mail: [email protected]

https://doi.org/10.1515/9783110571905-007

120 | H. Jafari et al.

1 Importance of groundwater and its modeling Water is an essential element for the development and maintenance of life on Earth. The main water resources on the Earth include surface water and subsurface water resources. Groundwater is defined as the portion of subsurface water that exists in the saturated zone [26]. Groundwater up to depths of 4000 m constitutes about 0.61 % of Earth’s total water and about 98 % of Earth’s available fresh water [1]. Groundwater is used for different purposes such as domestic, agricultural, and industrial demands. Globally, groundwater provides almost 50 % of the drinking water and almost 43 % of the irrigation water [39]. Groundwater is mainly found in aquifers, geological formations that store and transmit large amounts of water [40]. One kind of aquifers that hydrogeologists deal with are unconfined aquifers. The upper boundary of the unconfined aquifer is limited to a free groundwater surface where the pressure equals the atmospheric pressure. This free surface is called water table [15]. Most of extracting wells, qanats, subsurface drains of agricultural lands, etc., are examples of groundwater flow in an unconfined aquifer. The dynamic behavior of the groundwater level directly corresponds to the recharge and discharge of groundwater. The excess recharge of groundwater can cause environmental problems such as water logging of lands and soil salinity in regions with shallow groundwater. On the other hand, the excess discharge may lead to severe withdrawal of the water table, which causes failure of low-depth wells, subsidence, or saline water intrusion into groundwater in coastal aquifers [19, 37]. According to the abovementioned, the success of recharge and discharge projects of groundwater in a region, such as the construction of extracting wells and/or subsurface drainage systems, depends strongly on the groundwater level. Therefore, the development of mathematical models that can be applied to accurately simulate the dynamic behavior of groundwater level is of utmost importance. In addition, the accurate design of recharge and discharge systems requires knowledge of the governing differential equations on groundwater flow.

2 Governing integer partial differential equation on unconfined groundwater Groundwater always moves from the zones of higher hydraulic head to the zones of lower hydraulic head. Flow rate of groundwater (volume of water per unit time), Q (L3 T−1 ), is directly proportional to the cross-sectional area, A (L2 ), and the difference between hydraulic heads of two zones, h1 − h2 (L), and inversely proportional to the distance between two zones, L (L). Henry Darcy found these relationships in 1856 for the first time and the Darcy equation is the first mathematical model to describe

Fractional calculus for modeling unconfined groundwater

| 121

the groundwater flow. The algebraic and differential forms of Darcy’s equation are expressed as follows, respectively [6]: h1 − h2 , L dh Q = −Kx A , dx

Q = Kx A

(1) (2)

where Kx is the hydraulic conductivity of porous medium in the x-direction (LT−1 ) and dh is the hydraulic gradient in the x-direction (dimensionless). The Darcy equation dx can also be written as follows [6]: q = −Kx

dh , dx

(3)

where q = Q/A is called the specific discharge (LT−1 ). Equation (3) is the onedimensional form of Darcy’s equation, while the groundwater flow in nature may be three-dimensional. Therefore, the general form of Darcy’s equation is as follows [6]: q = −Kx

𝜕h 𝜕h 𝜕h − Ky − Kz . 𝜕x 𝜕y 𝜕z

(4)

The governing differential equation on groundwater flow in an unconfined aquifer with a horizontal impervious base is obtained by combining the Darcy equation and the principle of water mass conservation. This equation, which is also called Boussinesq equation (BE), is based on the validity of Darcy’s law and Dupuit assumptions. For further details of Dupuit assumptions, see Fetter [15]. The non-linear and linear forms of the BE are as follows, respectively: 𝜕 𝜕h 𝜕 𝜕h 𝜕h (K h ) + (Kx h ) + N = S , 𝜕x x 𝜕x 𝜕y 𝜕y 𝜕t 𝜕h 𝜕h 𝜕 N S 𝜕h 𝜕 (K ) + (Kx ) + = , 𝜕x x 𝜕x 𝜕y 𝜕y D D 𝜕t

(5) (6)

where N is the recharge rate or discharge rate (LT−1 ), S is the specific yield (dimensionless), D is the average saturated thickness (L), and t is the time (T). The BE has been solved using analytical (e. g., [14, 21, 36]), semi-analytical (e. g., [13]), and numerical (e. g., [4, 27, 11]) methods for various boundary and initial conditions. The main drawback of analytical and semi-analytical methods is that they assume the porous medium (aquifer) is homogeneous and the hydraulic conductivity is taken into account invariant to distance [36, 13]. However, the porous medium is heterogeneous in nature and the hydraulic conductivity varies with distance. This phenomenon is the so-called scale effect on the hydraulic conductivity [24]. One usual way to take the heterogeneity of porous media into account is the utilization of numerical methods to solve the BE. Although the numerical methods can consider the variation of porous medium properties [33], they suffer from conceptual errors, truncation

122 | H. Jafari et al. errors, data errors, instability under some conditions, a requirement of considerable computational capabilities, and long run-times [6, 5, 2]. The long run-time is the main drawback of numerical methods and it limits their application to practical projects [2, 22]. In recent years, fractional derivatives have been applied to remove or reduce the scale effects on the properties of porous media such as the dispersion coefficient and hydraulic conductivity. The reduction of scale effects by applying the fractional derivatives results from the non-locality characteristic of fractional derivatives [38]. This means that the fractional derivative of a function at a certain point not only depends on the value of the function at the point itself but also depends on the value of the function in the entire domain [24]. In this case, the scale effects are accounted for by the fractional differentiation order [9]. Huang et al. [18], Xiong et al. [43], and Gao et al. [16] compared the scale dependency of dispersion coefficients of the fractional advection-dispersion equation (FADE) and the advection-dispersion equation (ADE) in large homogeneous and heterogeneous saturated soil columns. According to their results, scale dependency of dispersion coefficients of the FADE is much less than that of the ADE. Application of the FADE in homogeneous and heterogeneous porous media, imitating homogeneous and heterogeneous unconfined aquifers, respectively, demonstrated that the FADE significantly reduced the scale effects on the dispersion coefficient [28]. In addition, according to the previous studies (e. g., [8, 32, 7, 25, 28]), the fractional equations of groundwater flow and solute transport in porous media are much more accurate than the classical equations, especially in the heterogeneous porous media.

3 Fractional governing partial differential equation on unconfined groundwater Various methods have been applied to derive the fractional governing equation on hydrologic processes such as fractional constitutive laws, probabilistic derivations, fractional mass conservation laws, and fractional mean value theorems [31]. In this section, the fractional governing partial differential equation on unconfined groundwater, which hereafter is called fractional Boussinesq equation (FBE), is derived using the fractional mass conservation law. To this end, from an unconfined aquifer that rests on a horizontal impervious bottom (Figure 1a), a control volume (Figure 1b) is considered. The mass conservation law for water flow through the control volume can be written as follows: 𝜕M ̇ + Δx) + M(y ̇ + Δy)) − (M(x) ̇ ̇ (M(x + M(y)) − ρNΔxΔy + = 0, 𝜕t

(7)

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| 123

Figure 1: Flow in an unconfined aquifer. (a) Schematic representation of an aquifer; (b) Control volume for flow through an unconfined aquifer with horizontal impervious bottom.

̇ + Δx) and M(y ̇ + Δy) are the water mass fluxes going out the control volume where M(x ̇ ̇ in the x- and y-directions, respectively (MT−1 ); M(x) and M(y) are the water mass flux going into the control volume in the x- and y-directions, respectively (MT−1 ); ρ is the ̇ + Δx) and M(y ̇ + Δy) are fluid density (ML−3 ) and M is the water mass (M). The M(x usually approximated using integer-order Taylor series as follows: ̇ 𝜕M(x) ̇ + Δx) = M(x) ̇ M(x + 𝜕x ̇ 𝜕 M(y) ̇ + Δy) = M(y) ̇ M(y + 𝜕y

̇ Δx 𝜕2 M(x) Δx 2 + + ⋅ ⋅ ⋅, ⋅ 1! 2! 𝜕x 2 ̇ Δy 𝜕2 M(y) Δy2 ⋅ + + ⋅ ⋅ ⋅. ⋅ 2 1! 2! 𝜕y ⋅

(8) (9)

̇ ̇ Using full Taylor series to approximate M(x+Δx) and M(y+Δy) results in an intractable mass conservation [42]. On the other hand, applying the truncated Taylor series (i. e., the first two terms) is only valid for linear or piece-wise linear variation of water mass flux [42]. To overcome the abovementioned limitations, one can assume the changes in water mass flux over the control volume obeying power-law functions with fractional orders and approximate them using the fractional Taylor series. Therefore, the variations of water mass flux within the control volume can be written as follows: ̇ + Δx) = M(x) ̇ ̇ ̇ M(x + Δxα ⇒ M(ϕ) = M(x) + (ϕ − x)α , ̇ + Δy) = M(y) ̇ ̇ ̇ M(y + Δyβ ⇒ M(φ) = M(y) + (φ − y)β ,

(10) (11)

where ϕ = x + Δx and φ = y + Δy are the auxiliary variables and α, β ∈ (0, 1]. According ̇ to the definition of the fractional Taylor series [30], one can approximate M(ϕ) at point x as follows: 󵄨󵄨 󵄨󵄨 ̇ ̇ 𝜕α M(ϕ) (ϕ − x)α Δx2α 𝜕α 𝜕α M(x) 󵄨󵄨 󵄨󵄨 ̇ ̇ M(ϕ) = M(x) + ⋅ ⋅ + ( ) + ⋅ ⋅ ⋅. (12) 󵄨 󵄨 󵄨 󵄨 󵄨󵄨ϕ=x Γ(1 + 2α) 𝜕ϕα 󵄨󵄨ϕ=x Γ(1 + α) 𝜕x α 𝜕x α Note that

̇ 𝜕α M(ϕ) 𝜕ϕα

̇ 𝜕α M(ϕ) 𝜕ϕα

is the Caputo fractional derivative of order 0 < α ≤ 1. Considering

= Γ(1 + α) and

̇ 𝜕α 𝜕α M(x) ( 𝜕xα ) 𝜕x α

̇ = 0, we can exactly obtain M(ϕ) using the first two

124 | H. Jafari et al. ̇ ̇ + terms of the fractional Taylor series. Therefore, the M(ϕ), which is the same as M(x Δx), can be exactly obtained as follows: 󵄨󵄨 ̇ 𝜕α M(ϕ) (ϕ − x)α 󵄨󵄨 ̇ ̇ M(ϕ) = M(x) + . ⋅ 󵄨󵄨 α 𝜕ϕ 󵄨󵄨ϕ=x Γ(1 + α)

(13)

One can write equation (13) in terms of x as follows: ̇ Δxα 𝜕α M(x) ̇ + Δx) = M(x) ̇ ⋅ . M(x + α 𝜕x Γ(1 + α)

(14)

̇ + Δy) can be exactly obtained as follows: Similarly, the M(y ̇ 𝜕β M(y) Δyβ ̇ + Δy) = M(y) ̇ . M(y + ⋅ Γ(1 + β) 𝜕yβ

(15)

By substituting equations (14) and (15) into equation (7), we get ̇ ̇ 𝜕α M(x) Δx α 𝜕β M(y) 𝜕M Δyβ ⋅ + − ρNΔxΔy + = 0. ⋅ β 𝜕x α Γ(1 + α) Γ(1 + β) 𝜕t 𝜕y

(16)

̇ ̇ The M(x) and M(y) are calculated as follows: ̇ M(x) = ρΔyhqx , ̇ M(y) = ρΔxhqy ,

M = ρSΔxΔyh.

(17) (18) (19)

By substituting equations (17), (18), and (19) into equation (16), we obtain 𝜕β (ρhqy ) Δyβ−1 𝜕α (ρhqx ) Δxα−1 𝜕(ρh) ⋅ + − ρN + S = 0. ⋅ 𝜕x α Γ(1 + α) Γ(1 + β) 𝜕t 𝜕yβ

(20)

In the unconfined aquifers, it is usually assumed that water is incompressible. Hence, equation (20) can be written as follows: 𝜕β (hqy ) Δyβ−1 𝜕α (hqx ) Δxα−1 𝜕h + −N +S = 0. ⋅ ⋅ α β 𝜕x Γ(1 + α) Γ(1 + β) 𝜕t 𝜕y

(21)

Equation (21) is the fractional water mass conservation equation in an unconfined resting on a horizontal impervious surface. By assuming the validity of Darcy’s law and Dupuit assumptions, one can derive the FBE by combining the fractional water mass conservation equation (equation (21)) and Darcy’s equation as follows: Δx α−1 𝜕h 𝜕α Δyβ−1 𝜕β 𝜕h 𝜕h ⋅ α (Kx h ) + ⋅ β (Ky h ) + N = S . Γ(1 + α) 𝜕x 𝜕x Γ(1 + β) 𝜕y 𝜕y 𝜕t

(22)

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| 125

As mentioned in the section 2, when the fractional derivatives are applied to model hydrogeology, the hydraulic parameters such as hydraulic conductivity are scaleinvariant and have constant values. Therefore, equation (22) can be written as follows: Kx

Δx α−1 𝜕h Δyβ−1 𝜕α 𝜕β 𝜕h 𝜕h ⋅ α (h ) + Ky ⋅ β (h ) + N = S . Γ(1 + α) 𝜕x 𝜕x Γ(1 + β) 𝜕y 𝜕y 𝜕t

(23)

β−1

α−1

Δy Δx and κy = Ky Γ(1+β) are taken into account, equation (23) will become as If κx = Kx Γ(1+α) follows:

κx

𝜕h 𝜕β 𝜕h 𝜕h 𝜕α (h ) + N = S , ) + κ (h y β 𝜕xα 𝜕x 𝜕y 𝜕t 𝜕y

(24)

where κx and κy have the dimensions (Lα T−1 ) and (Lβ T−1 ), respectively. Equation (24) is the non-linear form of the FBE for groundwater flow in an unconfined aquifer with horizontal impermeable bottom. The linear form of the FBE is obtained by assuming that the variation of h is infinitesimal in comparison with the saturated layer thickness. In this case, the average saturated thickness is equal to a constant value, D. Hence, the linear form of the FBE can be written as follows: κx

𝜕α 𝜕h 𝜕β 𝜕h N S 𝜕h = . ( ) + κy β ( ) + α 𝜕x 𝜕x D D 𝜕t 𝜕y 𝜕y

(25)

If we let μ = 1 + α and ν = 1 + β, the linear form of the FBE will become as follows: κx

𝜕μ h 𝜕ν h N S 𝜕h = , + κ + y 𝜕x μ 𝜕yν D D 𝜕t

(26)

where μ, ν ∈ (1, 2]. As observed, the FBE is a generalization of the BE, in which the first-order derivatives are replaced with the fractional-order derivatives. The FBE can model groundwater flow both in a heterogeneous aquifer and in a heterogeneous one as well as both in an anisotropic aquifer and in an isotropic one. The values of α and β are the indicators of heterogeneity degrees in the x- and y-directions, respectively. The smaller the values of α and β, the more heterogeneous the aquifer is, and vice versa. In addition, if α ≠ β and Kx ≠ Ky , the aquifer is anisotropic, while if α = β and Kx = Ky , the aquifer is isotropic. In a homogeneous and anisotropic aquifer, α = β = 1 and Kx ≠ Ky , while in a homogeneous and isotropic aquifer α = β = 1 and Kx = Ky .

4 Unsteady flow towards subsurface drains The unsteady flow towards subsurface drains (drainpipes) can be described by the BE [46]. Similarly, the FBE can be also applied to model the unsteady flow towards subsurface drains. In this section, the linear form of the FBE is analytically solved for

126 | H. Jafari et al.

Figure 2: Geometry and schematic representations of the drainage system considered in this chapter.

one-dimensional unsteady flow towards parallel subsurface drains. Figure 2 shows the schematic representation of the drainage system considered here. As shown in Figure 2, the distances and depths of drainpipes are equal and they lie above a horizontal impervious layer in a parallel manner. To solve the FBE for the drainage system indicated in Figure 2, the following assumptions are considered: 1. Flow towards the drainpipe is one-dimensional and horizontal. To compensate the error resulting from radial flow around the drainpipe, the Hooghoudt equivalent depth is used instead of the actual depth of the impervious layer below the drainpipe. 2. The water table rises suddenly due to an instantaneous recharge. After the recharge is stopped, the water table falls gradually and water is discharged by the drainpipe. 3. The water table at t = 0 is horizontal and the unsaturated flow above the water table is negligible. The linear form of the one-dimensional FBE is as follows: 𝜕h κx D 𝜕μ h = . 𝜕t S 𝜕x μ

(27)

According to the assumptions mentioned above, the initial and boundary conditions to solve the linear form of the one-dimensional FBE can be expressed as follows: h(x, 0) = h0 , h(0, t) = 0, h(L, t) = 0,

0 ≤ x ≤ L,

t > 0,

t > 0.

(28) (29) (30)

The FBE (equation (27)) is analytically solvable using the spectral representation approach [44]. Equation (27) can be written in terms of the fractional Laplacian operator as follows: κ D 𝜕h = − x (−Δ)μ/2 h. (31) 𝜕t S

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| 127

The spectral representation of equation (31) is presented based on Definition 2 in [44] on a finite domain [0, L] with homogeneous Dirichlet boundary conditions. The spectral representation of the Laplacian operator (−Δ) is obtained by solving the eigenvalue problem (λ) [44], i. e., − ΔH = λH,

H(0) = H(L) = 0.

(32)

The eigenvalues required in the spectral representation approach are considered as λn = ( nπ )2 for n = 1, 2, . . ., and the corresponding eigenfunctions are non-zero scalar L x). Next, the solution is given by multiples of H(x) = sin( nπ L ∞

h(x, t) = ∑ cn (t) sin( n=1

nπ x). L

(33)

As observed, equation (33) satisfies the boundary conditions mentioned in equations (29) and (30). Using Definition 2 in [44] and substituting h(x, t) into equation (31), we get dcn (t) κx D nπ + c (t)λnμ/2 } sin( x) = 0. dt S n L

∞

∑{

n=1

(34)

As observed, the problem for cn (t) is reduced to an ordinary differential equation, which has the following general solution: cn (t) = cn (0) exp(−

κx D μ/2 λ t). S n

(35)

The cn (0) is obtained using the initial condition (equation (28)). The initial condition can be rewritten as follows: ∞

h(x, 0) = h0 = ∑ cn (0) sin( n=1

nπ x). L

(36)

Equation (36) is a sine Fourier series of function h0 . The coefficient cn (0) is calculated as follows [10]: L

cn (0) =

2h 2 nπ ∫ h0 sin( θ)dθ = 0 (1 − cos(nπ)). L L nπ

(37)

0

)2 into equation (35), the cn (t) becomes as follows: Substituting cn (0) and λn = ( nπ L cn (t) =

μ

2h0 κ D nπ (1 − cos(nπ)) ⋅ exp(− x ( ) t). nπ S L

(38)

128 | H. Jafari et al. Therefore, the final solution of equation (27) subject to initial and boundary conditions mentioned in equations (28) to (30) is as follows: h(x, t) =

4h0 π

μ

κ D nπ 1 nπ exp(− x ( ) t) sin( x). n S L L n=1,3,5,... ∞

∑

(39)

Equation (39) simulates the water table profile between two parallel drainpipes which have equal installation depths. Equation (39), which hereafter is called the fractional Glover–Dumm equation (FGDE), is the generalization of the Glover–Dumm equation is replaced with the frac(GDE) [14] in which the second power of the expression nπ L tional order of μ (1 < μ ≤ 2). The only difference between the GDE and the FGDE is that the GDE assumes the porous medium is homogeneous [14], while the FGDE does not require this assumption. Therefore, the FGDE can be applied to both homogeneous and heterogeneous soils. The value of μ in the FGDE can be used as an indicator of the degree of heterogeneity of the porous medium so that the value of μ approaches 1 in the heterogeneous soil, while it approaches 2 in the homogeneous soil.

5 Application of the FGDE to laboratory and field data In this section, the FGDE is used to predict the water table profile between two drainpipes at laboratory and field scales, and its performance is compared to that of the GDE.

5.1 Laboratory tests The laboratory experiments were carried out in a sandbox with internal dimensions of 2 × 0.5 × 1.1 m (length × width × height). Figure 3 depicts a sketch of the laboratory setup. The sandbox was made of stainless steel. A piece of corrugated polyvinyl chloride (PVC) drainpipe with a length of about 0.5 m and an internal diameter of 100 mm was installed along one of the narrow ends of the sandbox. The drainpipe was installed at a height of 0.3 m above the bottom of the sandbox and wrapped with a synthetic envelope of PP450. A cut-off valve was installed at the outlet of the drainpipe to prevent water from exiting the drainpipe and three cut-off valves were installed at the bottom of the sandbox to saturate the soil from the bottom upward. The front plate of the sandbox included 11 manometers with a diameter of 1 cm at distances of 7, 22, 37, 52, 67, 86, 105, 124, 143, 162, and 184 cm from the drainpipe (Figure 3). The manometers were inserted up to the middle of the sandbox to remove the effect of seepage along the sandbox wall. The fluctuations of the water table were measured using these manometers during the laboratory tests.

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| 129

Figure 3: Sketch of the laboratory set-up.

The sandbox was filled to 1 m height with a relatively homogeneous sand of which the particle size varied in the range of 0.08 to 0.6 mm. Table 1 presents some physical properties of the sand used in the laboratory experiments. According to the value of the uniformity coefficient (Cu ), the used sand is uniform [41] and, consequently, is homogeneous [8]. Table 1: The physical properties of sand used. Actual soil density (g cm−3 )

Porosity (%)

d10 ∗ (mm)

d50 (mm)

d60 (mm)

Cu (dimensionless)

2.58

38.76

0.13

0.17

0.2

1.54

*

dxx denotes xx% of the sand particles are smaller than d.

The sand in the sandbox was saturated and drained several times to subside the sand. After subsiding the sand, the sandbox was re-filled with sand so that the final height of the sand in the sandbox was 1 m. The sand was saturated from the bottom upward by rising the water table gradually. To this end, an adjustable constant head reservoir was connected to the cut-off valves installed at the bottom of the sandbox. When the entire sand was saturated and water appeared on the sand surface, the cut-off valves were closed and the constant head reservoir was detached from the sandbox. Next, the water was drained by gradually opening the cut-off valves at the bottom of the sandbox. The cut-off valve installed at the outlet of the drainpipe was closed at this stage. After the last stage, the sand was saturated again from the bottom upward using the method mentioned above. After the sand was fully saturated, the cut-off valves at the bottom of the sandbox were closed and the constant head reservoir was disconnected from the sandbox. To collect data, the cut-off valve installed at the outlet of the drainpipe was opened and the values of the water tables were recorded using manometers at various times. The water table yield values, half of the water table profiles between

130 | H. Jafari et al. two parallel drainpipes, were measured at various times. It is necessary to mention that, due to symmetry, the other half of the profile is like the measured one. Also, the initial hydraulic head (h0 ) was obtained at 71 cm.

5.2 Field tests The final purpose of the mathematical drainage model is to predict the water table profile at field scale. Hence, in this subsection, the applicability of the FGDE at field scale is investigated. In this spirit, the measured data of water table profiles between two parallel drainpipes constructed in an experimental farm were used. The area of the experimental farm was 12 hectares and it was located in Abadan, Iran (30°1󸀠 49.8󸀠󸀠 N latitude and 48°29󸀠 5.27󸀠󸀠 E longitude). The soil texture varied from silt to clay and the average depth to the impervious layer was about 4.5 m. The drainage system of the experimental farm included the corrugated PVC drainpipes pre-wrapped with a synthetic envelope of PP450. The drainpipes had a length of 200 m and an internal diameter of 100 mm. The average installation depth and spacing of the drainpipes were 1.3 m and 30 m, respectively. To measure the water table depth, the observation wells equipped with perforated poly-ethylene tubes were installed at distances of 0, 0.5, 1.5, 5, 15, 25, 28.5, 29.5, and 30 m from the drainpipe. It is necessary to mention that the observed hydraulic head at each distance was obtained by subtracting the water table depth at that distance from the average drainpipe depth. The values of water table depths were measured daily during a period of 10 days.

5.3 Estimation of FGDE and GDE parameters In general, the mathematical models require some parameters to simulate a physical κ D process. The FGDE requires xS and μ to simulate the hydraulic head profile, while K D

the GDE needs Sx . The parameters of the FGDE and the GDE were estimated using the inverse problem method. To this end, two inverse models were developed. In the inverse models, the objective function (OF) was considered as follows: Minimize:

N

2

OF = ∑(hmeas − hsimu ), i i i=1

(40)

where hmeas and hsimu are the measured and simulated hydraulic heads at the ith i i point, respectively, and N is the number of measurement points. At the laboratory scale, the observed water table profiles at times t = 6, 16, 30, 45, 60, 90, 120, and 180 min after beginning drainage were used in the inverse models, while at the field scale, the profiles at times 2, 4, and 6 days after starting drainage were observed.

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| 131

To find the global optimum values of the FGDE and GDE parameters, the bees algorithm (BA) was used in the inverse models. The BA, first introduced by Pham et al. [34], is a population-based stochastic optimization algorithm. It imitates the foraging behavior of honey bees to solve an optimization problem [34]. The most important advantages of BA over other population-based stochastic optimization algorithm are [45, 23]: 1. its simplicity, flexibility, robustness, and non-sensitivity to the location of the global optimum solution at the search space; 2. the requirement of fewer control parameters and less sensitivity to configuration of control parameters; 3. easy tuning of control parameters for the various classes of problems and the ability to implement with mathematical operations; 4. it is easy to set the sampling rate around desirable areas of search space and applicability to various types of functions. Because of these advantages, the BA has been widely used to solve engineering optimization problems (e. g., [17, 29, 25, 3, 23]). The BA has been completely explained in the literature. Hence, here only the pseudo-code of the BA is presented [34]: 1. Initialize n random solutions. 2. Evaluate the fitness of the population using equation (40). 3. While (stopping criterion not met) forming new population. 4. Select the first m solutions as the best solutions and select them as the elite ones for neighborhood search. 5. Determine the patch size. 6. Produce ne new solutions around the elite solutions and ns ones around the best solutions so that ns < ne . 7. Select the fittest bee from each patch. 8. Produce the remaining n − m solutions randomly and evaluate their fitness. 9. End While.

Table 2: The best estimated values of FGDE and GDE parameters. Type of soil Homogeneous soil Experimental farm soil

Kx D S

(m2 day−1 ) 15.35 2.24

κx D S

(mμ day−1 )

μ (dimensionless)

16.25 0.18

1.99 1.10

Table 2 summarizes the obtained optimum values of the FGDE and GDE parameters using inverse models developed. As observed, the value of μ in the homogeneous soil is quite close to 2, while it is very close to 1 in the experimental farm soil, due to the heterogeneity of this soil. This means the value of μ in the FGDE can represent well

132 | H. Jafari et al. the heterogeneity degree of porous medium. Based on the classification proposed by Clarke et al. [12], the experimental farm soil is very heterogeneous. Also, the obtained values of FGDE and GDE parameters demonstrated that the FGDE reduces to the GDE in the homogeneous soil, which is in accordance with the theories mentioned in the section 4.

5.4 Comparison of FGDE and GDE to predict water table profile From a practical point of view, a mathematical drainage model must be able to predict the water table profiles at various times using the estimated parameters. Therefore, it is necessary to investigate the abilities of the FGDE and the GDE to simulate water table profiles at different times after beginning drainage. In this spirit, the water table profiles between two parallel drainpipes, both at laboratory and at field scales, were simulated for different periods of time not used in the stage of parameter estimation. To compare the performances of two models, statistical criteria and graphical displays were used. The statistical criteria included the error function (Ferror ), adjusted determination coefficient (R2adj ), maximum error (ME), and coefficient of residual mass (CRM), defined as follows [20, 35]: Ferror = √

N hmeas − hsimu 2 1 ), ⋅ ∑( i measi N − P i=1 hi

(41)

N −1 , N −P−1 󵄨󵄨 meas simu 󵄨󵄨 − hi 󵄨󵄨, ME = Max󵄨󵄨hi

(43)

CRM = 1 −

(44)

Radj = 1 − (1 − R2 ) ⋅

∑Ni=1 hsimu i , N meas ∑i=1 hi

(42)

where P is the number of parameters and R2 is the determination coefficient calculated as follows [35]: R2 = 1 −

− hsimu )2 ∑Ni=1 (hmeas i i , ∑N (hmeas − h̄ meas )2 i=1

i

(45)

where h̄ meas is the average of the measured data. Note that when the values of Ferror , R2adj , ME, and CRM approach 0, 1, 0, and 0, respectively, this implies that the model predicts the studied phenomenon well. The graphical displays applied to evaluate the FGDE and the GDE included: (i) a comparison of coincidence rate of water table profiles predicted by the FGDE and the GDE with those measured in the laboratory and the field; and (ii) a comparison of the match between predicted and observed values of hydraulic heads [20].

Fractional calculus for modeling unconfined groundwater

| 133

Table 3: The statistical criteria values as performance indicators of FGDE and GDE. Type of soil

GDE Ferror

2 Radj

ME

CRM

FGDE Ferror

2 Radj

ME (m)

CRM

Homogeneous soil Experimental farm soil

0.11 0.76

0.99 −1.19

0.07 0.67

−0.02 0.37

0.11 0.57

0.99 0.64

0.07 0.22

−0.02 0.12

Table 3 lists the values of statistical criteria corresponding to the FGDE and the GDE both for homogeneous soil and for experimental farm soil. As shown in Table 3, the values of statistical criteria of two models are very close to each other in the homogeneous soil. This means that the FGDE reduces to the GDE in the homogeneous porous media. The values of statistical criteria of the FGDE and the GDE demonstrate that both models can well simulate water table profiles in the homogeneous soil. The graphical displays also confirm that the performances of the FGDE and the GDE are similar in the homogeneous soil (Figures 4 and 5).

Figure 4: Observed and simulated water table profiles at different times after beginning drainage.

In contrast to the homogeneous soil, the performances of the FGDE and the GDE are completely different in the experimental farm soil. The corresponding values of statistical criteria (Table 3) demonstrate that, compared to the GDE, the FGDE better simulates the values of hydraulic heads. The graphical comparison of the FGDE and GDE performances also reflects the superiority of the FGDE over the GDE (Figures 6 and 7). The negative value of R2adj related to the GDE implies that the values of the hydraulic heads predicted by the GDE are worse than the average of the measured values [20]. Also, according to the values of CRM and Figure 7, the GDE strongly under-predicts the values of hydraulic heads. The weaker performance of the GDE results from the homogeneity assumption considered to derive this model. In contrast to the GDE, the FGDE takes into account the heterogeneity of porous medium as a determinable pa-

134 | H. Jafari et al.

Figure 5: Measured versus simulated hydraulic heads above the drainpipe. (a) GDE; (b) FGDE. The line represents the potential 1:1 relationship between measured and simulated data.

Figure 6: Observed and simulated water table profiles at times (a) t = 3 days and (b) t = 10 days after beginning drainage.

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| 135

Figure 7: Measured versus simulated hydraulic heads above drainpipe. (a) GDE; (b) FGDE. The line represents the potential 1:1 relationship between measured and simulated data.

rameter of which the value varies for different porous media. Therefore, the FGDE has fewer limitations and, consequently, better performance.

6 Concluding remarks In this chapter, the fractional governing partial differential equation on unconfined groundwater, which was also called the FBE, was derived using the fractional mass conservation law. The FBE can be applied to describe the groundwater flow both in homogeneous and in heterogeneous porous media. In addition to hydrodynamic coefficients (K and S), the FBE includes another parameter, which is the indicator of the degree of porous medium heterogeneity. This indicator is in the range of 0 to 1 in the non-linear form of the FBE. As the value of the heterogeneity degree indicator approaches 0, the porous medium is very heterogeneous, and vice versa. When the value of the heterogeneity degree indicator is equal to 1, the FBE reduces to the BE.

136 | H. Jafari et al. The linear form of the FBE was analytically solved for one-dimensional unsteady flow towards parallel subsurface drains. The resultant solution, called the FGDE, was fitted to the laboratory and field data of water table profiles between two drainpipes, and its performance was compared to that of the GDE. The parameters of the FGDE and the GDE were estimated using the inverse problem method. The results indicate that the obtained values for the indicator of the degree of heterogeneity accurately reflected the heterogeneity degree of the soil examined. Comparison of the FGDE and the GDE demonstrated that the two models had similar performances in the homogeneous soil. In other words, the FGDE reduces to the GDE in the homogeneous soil. However, the performances of the FGDE and the GDE were completely different in the heterogeneous soil. In this soil, the FGDE much better simulated water table profiles as compared to the GDE. In summary, the FBE shows significant potential for describing groundwater flow in unconfined aquifers with various degrees of heterogeneity.

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Yury Rossikhin and Marina Shitikova

Fractional calculus models in dynamic problems of viscoelasticity Abstract: In the present chapter, viscoelastic operators are constructed for the fractional derivative Kelvin–Voigt, Maxwell, and standard linear solid models with and without volume relaxation involving time-dependent Poisson’s operators, since these models are frequently used in engineering applications. The suitability of these models for real advanced materials including auxetics is evaluated. Keywords: Viscoelasticity, Riemann–Liouville fractional derivative, Rabotnov fractional exponential function, viscoelastic fractional operators, time-dependent Poisson’s operator MSC 2010: 65C05, 62M20, 93E11, 62F15, 86A22

1 Preliminary remarks It is well known that each isotropic elastic material possesses only two independent constants, and all others are expressed in terms of two constants which should be preassigned [11]. Thus, if Young’s modulus E and Poisson’s ratio ν are known, then Lame constants λ and μ and bulk modulus K are expressed as μ=

E , 2(1 + ν)

λ=

Eν , (1 − 2ν)(1 + ν)

K=

E , 3(1 − 2ν)

(1)

or, in the case when shear modulus μ and bulk modulus K are pre-assigned, other constants, E, ν, and λ, are defined as E=

9Kμ , 3K + μ

ν=

3K − 2μ , 2(3K + μ)

λ=

3K − 2μ . 9

(2)

In the case of an isotropic viscoelastic material, material properties are timedependent, and once again only two time-dependent viscoelastic operators should be known, while others could be expressed in terms of two pre-assigned operators utilizing the correspondence principle and relationship (1) or (2). Acknowledgement: This work is supported by the Russian Foundation for Basic Research, Project No. 17-01-00490. Yury Rossikhin, Marina Shitikova, Research Center on Dynamics of Solids and Structures, Voronezh State Technical University, 20-letiya Oktyabrya Str. 84, Voronezh 394006, Russian Federation, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110571905-008

140 | Y. Rossikhin and M. Shitikova It has been emphasized by Rabotnov [18], who was one of the pioneers in the application of fractional operators based on the fractional derivatives in mechanics of solids, that the majority of experiments carried out with viscoelastic materials are creep experiments; that is why if a rheological model is written in the form of the stress-strain relationship, then it is necessary to find the reverse connection, i. e., to express the strain ε in terms of the stress σ. Precisely this relation allows one to carry out creep experiments and to determine the physical constants involved in this strainstress relationship. In other words, it is a need to construct the resolvent operators for each model. In the present chapter, viscoelastic operators will be constructed for the fractional derivative Kelvin–Voigt, Maxwell, and standard linear solid models with and without volume relaxation, since these models are frequently used in engineering applications [21]. The suitability of these models for real advanced materials will be evaluated.

2 The simplest fractional derivative viscoelastic models Let us consider the simplest viscoelastic fractional calculus models which could be obtained by substituting the integer-order derivatives in the conventional models of viscoelasticity by the fractional-order time derivatives using the Riemann–Liouville definition [29] RL γ

D σ=

t

d σ(t 󸀠 ) dt 󸀠 , ∫ dt Γ(1 − γ)(t − t 󸀠 )γ

(3)

0

where 0 < γ ≤ 1 is the order of the fractional derivative and Γ(1 − γ) is the Gammafunction. It should be noted that the choice of the Riemann–Liouville derivative of the fractional order for studying all problems within this chapter is not random. Since during the solution of dynamic problems of viscoelasticity it is often needed to find resolvent operators and to decode intricate operator relationships [22, 24], the utilization of other definitions of fractional derivatives, the Gerasimov–Caputo derivative GC Dγ as an example [6, 3] turns out to be inconvenient. Thus, in Diethelm [5] it has been proven that “the Caputo derivative is a left inverse of the Riemann–Liouville integral I γ , . . . but the Caputo derivative is not the right inverse of the Riemann–Liouville integral”, i. e., C Dγ I γ ≠ I γ C Dγ , despite the Riemann–Liouville fractional derivative which provides RL Dγ I γ = I γ RL Dγ . In other words, the inequality due to the Caputo derivative results in additional difficulties when finding resolvent operators. That is why in the further treatment we will utilize fractional derivatives in the Riemann–Liouville sense.

Fractional calculus models in dynamic problems of viscoelasticity | 141

2.1 Fractional derivative Kelvin–Voigt model The fractional derivative Kelvin–Voigt model was introduced by Shermergor in 1966 [30] and it has the form σ = E0 ε + E0 τσγ Dγ ε,

(4)

where τσ is the retardation (creep) time and E0 is the relaxed elastic modulus (prolonged modulus of elasticity, or the rubbery modulus). At γ = 1, the model (4) goes over into a conventional Kelvin–Voigt model of viscoelasticity. The equation resolvent to (4) has the form ε(t) = J0

1 σ(t), γ 1 + τσ Dγ

(5)

where J0 = E0−1 is the prolonged compliance, while the operator 1 γ 1 + τσ Dγ

∋∗γ (τσγ ) =

(6)

is the dimensionless Rabotnov operator [22, 23, 24]. Considering that Dγ I γ = I γ Dγ = 1, the operator (6) could be represented as ∋∗γ (τσγ ) =

I γ τσ

−γ

1 − (−I γ τσ ) −γ

,

(7)

where γ

t

I σ=∫ 0

(t − t 󸀠 )γ−1 σ(t 󸀠 )dt 󸀠 Γ(γ)

(8)

is the fractional integral. If it is supposed that the right part of equation (7) is the sum of an infinite decreas−γ ing geometrical progression, the denominator of which is equal to d = −I γ τσ , then it follows that ∞

∋∗γ (τσγ ) = ∑ (−1)n τσ−γ(n+1) I γ(n+1) . n=0

(9)

Considering (9), the strain-stress relationship (5) could be rewritten as ε=

J0 ∋∗γ (τσγ )σ(t)

t

= J0 ∫ ∋γ (t 󸀠 /τσ )σ(t − t 󸀠 )dt 󸀠 ,

(10)

0

where ∋γ (t/τσ ) =

t γ−1 ∞ (−1)n (t/τσ )γn γ ∑ τσ n=0 Γ[γ(n + 1)]

(11)

142 | Y. Rossikhin and M. Shitikova is the fractional exponential function suggested by Rabotnov in 1948 [17] which at γ = 1 goes over into a conventional exponential function. But when γ → 0, it transforms into the δ-like sequence, since ∋0 (t/τσ ) vanishes to zero at any t (Γ(0) = ∞) except the magnitude t = 0, at which ∋0 (0) = ∞, i. e., 1 lim ∋γ (t 󸀠 /τσ ) = δ(t 󸀠 ). 2

γ→0

(12)

That is why at γ = 0, as it follows from (10), 1 ε(t) = J0 σ(t), 2

(13)

while from the stress-strain relationship (4) at γ = 0 we have σ(t) = 2E0 ε(t).

(14)

2.2 Fractional derivative Maxwell model The fractional derivative Maxwell model has the form J∞ (σ + τεγ Dγ σ) = τεγ Dγ ε,

(15)

−1 where J∞ = E∞ is the instantaneous compliance, E∞ is the non-relaxed (instantaneous, or glassy) modulus of elasticity, and τε is the relaxation time. The stress-strain relationship (15) could be rewritten in another form if we apply the operator I γ to the left- and right-hand sides of (15) and consider that I γ Dγ = Dγ I γ = 1. As a result we obtain the strain-stress relationship corresponding to the generalized Maxwell model written in terms of the fractional integral, which was for the first time introduced by Shermergor in 1966 [30], i. e., t

ε(t) = J∞ [σ(t) +

τε−γ

∫ 0

t 󸀠 γ−1 σ(t − t 󸀠 )dt 󸀠 ]. Γ(γ)

(16)

Expressing the value of σ(t) from (15) yields σ(t) = E∞

γ

τε Dγ 1 )ε(t). γ γ ε(t) = E∞ (1 − γ 1 + τε D 1 + τε Dγ

(17)

Considering (6) and (9), relationship (17) could be rewritten in the following form: t

σ(t) = E∞ [ε(t) − ∫ ∋γ (t 󸀠 /τε )ε(t − t 󸀠 )dt 󸀠 ]. 0

(18)

Fractional calculus models in dynamic problems of viscoelasticity | 143

If γ tends to 0 in (16) and (18) and we consider t 󸀠 γ−1 = δ(t 󸀠 ), γ→0 Γ(γ) 1 lim ∋γ (t 󸀠 /τε ) = δ(t 󸀠 ), γ→0 2 lim

(19) (20)

it can be found that ε(t) = 2J∞ σ(t), 1 σ(t) = E∞ ε(t). 2

(21) (22)

2.3 Fractional derivative standard linear solid model The fractional derivative standard linear solid model was for the first time suggested by Meshkov in 1967 [13], i. e., J0 (σ + τεγ Dγ σ) = ε + τσγ Dγ ε, where

γ

(

J τε E ) = 0 = ∞. τσ E∞ J0

(23)

(24)

It should be noted that equation (24), which was derived in Shermergor [30] and Meshkov [13] from the comparison of the resolvent operators describing the stressstrain and strain-stress relationships, is very important from the physical point of view, since it provides the coupling between the rheological parameters of the model to ensure its physical validity [15]. Let us first express σ(t) in terms of ε(t) from (23). As a result we obtain σ(t) = E0

γ

1 + τσ Dγ ε(t), γ 1 + τε Dγ

(25)

or with due account for (24) σ(t) = E0

γ

1 + E∞ E0−1 τε Dγ γ

1 + τε Dγ

ε(t),

(26)

which could be reduced to σ(t) = E∞ [1 − νε

1 ∗ γ γ γ ]ε(t) = E∞ [1 − νε ∋γ (τε )]ε(t), 1 + τε D

(27)

−1 where νε = △EE∞ and △E = E∞ − E0 is the defect of the modulus, i. e., the value characterizing the decrease in the elastic modulus from its non-relaxed value to its relaxed value.

144 | Y. Rossikhin and M. Shitikova Considering (6) and (9), relationship (27) could be rewritten in the following form: t

σ(t) = E∞ [ε(t) − νε ∫ ∋γ (t 󸀠 /τε )ε(t − t 󸀠 )dt 󸀠 ].

(28)

0

in

Then the strain ε(t) could be expressed in terms of stress σ(t) from (23), resulting

ε(t) = J0

γ

1 + E∞ E0−1 τε Dγ 1+

γ τσ Dγ

σ(t),

(29)

σ(t),

(30)

or with due account for (24) ε(t) = J0

γ

1 + J∞ J0−1 τσ Dγ γ

1 + τσ Dγ

which could be reduced to ε(t) = J∞ [1 + νσ

1 ]σ(t) = J∞ [1 + νε ∋∗γ (τσγ )]σ(t), γ 1 + τσ Dγ

(31)

−1 and △J = J0 − J∞ . where νσ = △JJ∞ Considering (6) and (9), relationship (31) could be rewritten in the following form: t

ε(t) = J∞ [σ(t) + νσ ∫ ∋γ (t 󸀠 /τσ )σ(t − t 󸀠 )dt 󸀠 ].

(32)

0

At γ → 0, expressions (28) and (32) take, respectively, the forms 1 σ(t) = E∞ (1 − νε )ε(t), 2 1 ε(t) = J∞ (1 + νσ )σ(t). 2

(33) (34)

Considering that −1 1 − νε = E0 E∞ = τεγ τσ−γ ,

−1 1 + νσ = J0 J∞ = τσγ τε−γ ,

(35)

from relationships (33) and (34) we have 1 σ(t) = (E∞ + E0 )ε(t) = Eε(t), 2 1 ε(t) = (J∞ + J0 )σ(t) = Jσ(t). 2 But, as seen from equation (24), E = E0 = E∞ ,

J = J0 = J∞ ,

(36) (37)

Fractional calculus models in dynamic problems of viscoelasticity | 145

and hence at γ = 0 the fractional derivative standard linear solid model goes over into the correct model of a pure elastic body. Based on equations (28) and (32), it is possible to write the connection between the resolvent operators, i. e., 1 ∗ γ γ = 1 + νσ ∋γ (τσ ). 1 − νε ∋∗γ (τε )

(38)

In order to prove the relationship (38), it is necessary to multiply its right-hand side by the denominator of the fraction in its left-hand side, to apply the theorem of multiplication of Rabotnov’s operators [22, 24], i. e., ∋∗γ (τεγ )∋∗γ (τσγ ) =

γ

γ

γ

γ

τε ∋∗γ (τε ) − τσ ∋∗γ (τσ ) γ

γ

τε − τσ

,

(39)

as well as to consider the formulae γ

γ

νσ τε νε τσ γ γ = γ γ = 1. τσ − τε τσ − τε

2.4 Koeller model and the generalized Rabotnov model The Koeller model [10] is the immediate generalization of the fractional derivative standard linear solid model (23) by involving in the left and right sides by n fractional time derivative terms, instead of two terms in the model (23), i. e., n

n

i=0

j=0

∑ ai Diγ ε = ∑ bj Djγ σ,

(40)

where Diγ and Djγ are the Riemann–Liouville derivatives (3) and ai and bj are some coefficients. Expressing the strain ε in terms of the stress σ from (40) and vice versa, we have ε= σ=

∑nj=0 bj Djγ

σ,

(41)

∑ni=0 ai Diγ ε. ∑nj=0 bj Djγ

(42)

∑ni=0 ai Diγ

Suppose that equations n

∑ bj Z j = 0,

j=0 n

∑ ai Y i = 0

i=0

(43) (44)

146 | Y. Rossikhin and M. Shitikova possess only simple real negative roots Zj = −tj (j = 1, . . . , n) and Yi = −τi (i = 1, . . . , n). Then dividing in (41) and (42) the polynomials standing in the numerators by those in the denominators, and further decomposing the proper fractions obtained in the remainder into simple fractions with due account for the assumptions for the roots of (43) and (44), we have −γ

−γ

n

γ

(45)

σ = E∞ [1 − ∑ mj ∋∗γ (tj )]ε, j=1 n

γ

(46)

ε = J∞ [1 + ∑ ni ∋∗γ (τi )]σ, i=1

γ

γ

−1 where J∞ = bn a−1 n , E∞ = an bn , τi (i = 1, 2, . . . , n) are retardation times, tj (j = 1, 2, . . . , n)

are relaxation times, ni =

γ gi τi

and mj =

γ ej tj

are constants, and

−1

n−1

n b a −γ −kγ −γ gi = ∑ ( k − k )τi (−1)k [∏(τl − τi )] , b a n n l=1 k=0 (l=i) ̸

−1

n a b −γ −γ −kγ ej = ∑ ( k − k )tj (−1)k [∏(tl − tj )] . b a n n l=1 k=0 n−1

(l=j) ̸

Moreover, it could be shown that the models (45) and (46) are resolvent only if the following equalities are valid: n

1+∑

mj tj

−γ

j=1

τi − tj

n

ni τi

1+∑ i=1

−γ

−γ

−γ

τi − tj −γ

−γ

= 0,

(47)

= 0.

(48)

From the nth-order equations (48) we could define n magnitudes of tj

−γ

(j =

1, . . . , n), while knowing tj from the set of n equations (47) we find the values mj (j = 1, . . . , n). −γ We suppose now that constants mj and tj (j = 1, . . . , n) are known, and we need −γ

to determine constants ni and τi (i = 1, . . . , n). In this case, from the nth-order equa−γ −γ tions (47) we could define n magnitudes of τi , while knowing τi , we could find the values of ni from the set of n equations (48). The analysis of relationships (47) and (48) shows that the following constraints are implied on the relaxation and retardation times [22]: −γ

τk < tk < τk+1 , −γ

−γ

−γ

τn−γ < tn−γ .

(49)

Note that relationships (47) and (48) differ a little from those presented in Rabotnov [18, 23], since in this chapter we use fractional operators in dimensionless form,

Fractional calculus models in dynamic problems of viscoelasticity | 147

which has allowed us to generalize relationship (24) for the case of the generalized Rabotnov model [22, 23], i. e., γ

( (

n ∏ni=1 τi ) = 1 + ∑ ni , n ∏j=1 tj i=1

(50)

) = 1 − ∑ mj .

(51)

∏nj=1 tj

∏ni=1 τi

γ

n

j=1

Equations (50) and (51) are the immediate extension of equations (35). To prove equation (50), we adopt relationship (48) rewritten in the following form: n

1+∑ i=1

ni γ = 0, 1 − xτi

(52)

where x = t −γ . Reducing all terms of (50) to the common denominator, we are led to the equation of the nth order, i. e., xn + c1 xn−1 + c2 xn−2 + ⋅ ⋅ ⋅ + cn = 0, where cn =

1 + ∑ni=1 ni

γ.

(−1)n ∏ni=1 τi

(53)

(54)

Utilizing one of the Viet formulae concerning the roots of the algebraic nth-order equation, i. e., n

cn = (−1)n ∏ tj , −γ

j=1

(55)

and substituting cn by (55), we are led to the relationship (50). In a similar way, to prove equation (51) we adopt relationship (47) rewritten in the following form: n

1−∑ j=1

mj

γ

1 − ytj

= 0,

(56)

where y = τ−γ . Reducing all terms of (56) to the common denominator, we are led to the equation of the nth order similar to (53), where cn =

1 − ∑nj=1 mj

γ.

(−1)n ∏nj=1 tj

(57)

148 | Y. Rossikhin and M. Shitikova

tion

Considering the Viet formula concerning the roots of the algebraic nth-order equan

cn = (−1)n ∏ τi , i=1

−γ

(58)

and equating relationships (57) and (58) to each other, we are led to equation (51). If γ tends to 0 in (50) and (51), this yields, respectively, n

∑ ni = 0 i=1

and

n

∑ mj = 0. j=1

(59)

(60)

If now the fractional parameter γ → 0 in the generalized Rabotnov resolvent models (45) and (46) with due account for (20), (59), and (60), we are led to two unrepugnant equalities (36) and (37) describing the pure elastic behavior of the material.

3 Application of the fractional derivative viscoelastic models for the description of dynamic response of current materials During the description of such viscoelastic bodies as beams, plates, and shells, the most frequently used models for the Young operator are the fractional derivative Kelvin–Voigt relation (4), the fractional derivative Maxwell model (15), and the fractional derivative standard linear solid model (23). In doing so the Poisson’s ratio of viscoelastic material in the majority of the cases is assumed to be a constant [21, 22]. However, as experimental data have shown [9, 18], the Poisson’s ratio is always a time-dependent operator ν̃ [7, 8, 32], and only the bulk extension-compression operã may be expressed as the time-independent value, which for the most viscoelastic tor K materials weakly varies during deformation. But the fractional derivative Kelvin–Voigt model with a time-independent Poisson’s ratio is only acceptable for the description of the dynamic behavior of elastic bodies in a viscoelastic medium [12, 21, 26, 28] or on a viscoelastic foundation [1, 31]. The review of “traditional” fractional calculus models (“traditional” in the sense that such models consider time-independent Poisson’s ratios) could be found in [20, 19, 21]. Below we will overview the fractional derivative models involving the timedependent Poisson’s operators, what allow one to reveal rather interesting features of

Fractional calculus models in dynamic problems of viscoelasticity | 149

advanced viscoelastic materials, among them auxetic materials possessing negative Poisson’s ratios [2, 4].

3.1 Modeling the Young operator Ẽ using the fractional derivative Kelvin–Voigt model without volumetric relaxation In order to show insolvency of the fractional derivative Kelvin–Voigt model, following Rossikhin et al. [27], let us choose Young’s operator as γ Ẽ = E0 [1 + (τσE ) Dγ ],

(61)

where τσE is the retardation, or creep, time during longitudinal deformations and Dγ is the Riemann–Liouville fractional derivative (3). ̃ is assumed to be time-independent, The bulk extension-compression operator K i. e., volumetric relaxation is neglected (this assumption is due to the fact that for many viscoelastic materials volumetric relaxation is much smaller than the shear relaxation), i. e., ̃ = K0 , K

(62)

Ẽ = 3K0 , 1 − 2ν̃

(63)

resulting in

where K0 is a certain constant. Substituting (61) in (63), we could find Poisson’s operator ν̃ = ν0 −

E0 E γ γ (τ ) D , 6K0 σ

(64)

and finally ν(t) = ν̃H(t) = ν0 −

E0 E γ t −γ (τ ) , 6K0 σ Γ(1 − γ)

(65)

lim ν̃H(t) = ν0 ,

(66)

whence it follows that lim ν̃H(t) = −∞,

t→0 3K −E

t→∞

0 0 . where ν0 = 6K 0 From (66) it is seen that this model lacks a physical meaning, since for real materials the low limiting value of Poisson’s ratio ν(0) could not take on the extremely negative value −∞. Thus this model is inappropriate for dealing with real viscoelastic materials.

150 | Y. Rossikhin and M. Shitikova

̃ using the fractional derivative 3.2 Modeling the shear operator μ Kelvin–Voigt model without volumetric relaxation ̃ is most frequently pre-assigned using the fractional derivative The shear operator μ Kelvin–Voigt model γ

̃ = μ0 [1 + (τσμ ) Dγ ], μ

(67)

μ

where μ0 is the relaxed shear modulus, τσ is the retardation time during shear deformations, while the bulk operator is assumed to be constant according to (62). In order to evaluate the dynamic response of viscoelastic bodies, it is necessary to calculate Young’s operator. For this purpose using the Volterra correspondence principle, the following formula could be utilized: ̃ 9K0 μ Ẽ = . ̃ 3K0 + μ

(68)

̃ = (3K0 + μ0 )(1 + tσγ Dγ ), 3K0 + μ

(69)

First we write the operator

γ

μ

where tσ = μ0 (τσ )γ (3K0 + μ0 )−1 . Then we find the operator reverse to (69), i. e., ̃ )−1 = (3K0 + μ

1 1 = (3K0 + μ0 )−1 ∋∗γ (tσγ ). 3K0 + μ0 1 + tσγ Dγ

(70)

Substituting (67) and (70) in (68) and considering one useful formula [27] tσγ Dγ ⋅ ∋∗γ (tσγ ) =

γ

tσ Dγ = 1 − ∋∗γ (tσγ ), γ 1 + tσ Dγ

(71)

this yields E Ẽ = 9K0 [1 − 0 ∋∗γ (tσγ )]. 3μ0

(72)

Now Poisson’s operator ν̃ could be calculated via the formula Ẽ = 3K0 . 1 − 2ν̃

(73)

Substituting (72) in (73), we have ν̃ = −1 +

E0 ∗ γ ∋ (t ). 2μ0 γ σ

(74)

Fractional calculus models in dynamic problems of viscoelasticity | 151

If we study the relaxation process, i. e., consider the longitudinal deformation in ̃ a rod as constant, then the operator Ẽ will act on the unit Heaviside function EH(t). With due account for ∞

∫ ∋γ (− 0

t − t󸀠 )H(t 󸀠 )dt 󸀠 = 1, τi

(75)

we obtain ̃ lim EH(t) = 9K0 ,

t→0

9K0 μ0 ̃ = lim EH(t) = E0 . 3K0 + μ0

(76)

3K0 − 2μ0 = ν0 . 2(3K0 + μ0 )

(77)

t→∞

For operator ν̃H(t) we have [27] lim ν̃H(t) = −1,

t→0

lim ν̃H(t) =

t→∞

According to the classical theory of elasticity it has been considered that for the conventional materials the Poisson’s ratio ν varies within the interval 0 < ν ≤ 0.5. However, nowadays a wide variety of so-called auxetic materials has been fabricated, including polymeric and metallic foams, micro-porous polymers, carbon fiber laminates, and honeycomb structures [2, 4], which possess unusual mechanical properties such as a negative Poisson’s ratio, i. e., the Poisson’s ratio ν could vary within the iñ = K0 = const could describe the terval −1 < ν ≤ 0.5. Therefore, the model (67) with K behavior of viscoelastic auxetic materials, and in so doing the Poisson’s ratio could vary from −1 to its relaxed magnitude ν0 .

̃ using the fractional derivative 3.3 Modeling the shear operator μ Maxwell model If the fractional derivative Maxwell model is applied for describing viscoelastic bodies, ̃ could be written in the form then the shear operator μ ̃ = μ∞ τεγ Dγ ⋅ ∋∗γ (τεγ ), μ

(78)

or, using equation (71), in the form ̃ = μ∞ [1 − ∋∗γ (τεγ )], μ

(79)

where μ∞ is the non-relaxed shear modulus and τε is the relaxation time. In doing so the volumetric operator is still considered as a constant, i. e., ̃ = K∞ . K

(80)

152 | Y. Rossikhin and M. Shitikova Using the procedure described above for the Kelvin–Voigt model, we could similarly obtain for the Maxwell model Ẽ = E∞ [1 − ∋∗γ (τσγ )], ν̃ =

E 1 − ∞ [1 − ∋∗γ (τεγ )], 2 6K∞

(81) (82)

or ν̃ = ν∞ +

E∞ ∗ γ ∋ (τ ), 6K∞ γ ε

(83)

where ν∞ =

3K∞ − μ∞ . 2(3K∞ + μ∞ )

If we consider the relaxation process, then t→0

̃ lim EH(t) = E∞ ,

t→∞

lim ν̃H(t) = ν∞ ,

t→∞

t→0

̃ lim EH(t) = 0,

(84)

1 lim ν̃H(t) = . 2

(85)

From (85) it is evident that for the fractional derivative Maxwell model, Poisson’s ratio could increase from ν∞ to its limiting magnitude of 1/2, which means that this model is suitable for the analysis of viscoelastic rubber-like materials.

̃ using the fractional derivative 3.4 Modeling the shear operator μ standard linear solid model If the fractional derivative standard linear solid model is applied for describing vis̃ has the form coelastic bodies, then the shear operator μ ̃ = μ0 μ

γ

1 + τσ Dγ , γ 1 + τε Dγ

(86)

or ̃ = μ0 ∋∗γ (τεγ ) + μ0 μ

γ

τσ γ γ ∗ γ γ τε D ⋅ ∋γ (τε ). τε

(87)

Considering equation (71) in (87) and introducing the notation μ∞ = μ0

γ

τσ γ, τσ

(88)

Fractional calculus models in dynamic problems of viscoelasticity | 153

as a result we obtain [27] ̃ = μ∞ [1 − νμε ∋∗γ (τεγ )], μ

(89)

̃ where νμε = (μ∞ − μ0 )μ−1 ∞ . In doing so the operator K is still defined by equation (80). For this model, the following relationships are valid: 3K Ẽ = E∞ [1 − Mσ ∞ ∋∗γ (τσγ )], μ∞ E∞ Mσ ∗ γ ν̃ = ν∞ + ∋ (τ ), 2μ∞ γ σ

(90) (91)

where 1 − μ0 μ−1 3K∞ ∞ = < 1, μ∞ 1 + μ0 (3K∞ )−1 E∞ Mσ 1 < 1. = 2μ∞ 2K∞ (1 + μ∞ /3K∞ )(1 + μ0 /3K∞ ) Mσ

Reference to relationships (89), (90), and (91) shows that the fractional derivative standard linear solid model fits perfectly not only for describing the behavior of conventional viscoelastic materials [27], but also for considering the changes in microstructure of the viscoelastic material as a result of external mechanical or physical impacts [16].

3.5 Scott Blair model for shear relaxation Some authors prefer to use the simplest fractional derivative model, i. e., the Scott Blair element, for modeling the shear operator ̃ = μτγ Dγ , μ

(92)

̃ = K = const. and assume that volumetric relaxation is absent, i. e., K In this case, first we could calculate ̃+μ ̃ = 3K + μτγ Dγ = 3K(1 + 3K

μ γ γ τ D ) = 3K(1 + T γ Dγ ), 3K

(93)

μ

where T γ = 3K τγ . Then considering equation (6), the viscoelastic Poisson’s operator for this case could be found easily in the form ν̃ =

̃ − 2μ ̃ 3K ̃ − 2μτγ Dγ ) 1 ∋∗ (T γ ) = −1 + 3 ∋∗ (T γ ), = (3K ̃ 6K γ 2 γ ̃) 2(3K + μ

(94)

154 | Y. Rossikhin and M. Shitikova whence it follows that 3 ν(t) = ν̃H(t) = −H(t) + ∋∗γ (T γ )H(t), 2 1 lim ν̃H(t) = −1, lim ν̃H(t) = . t→∞ t→0 2

(95) (96)

From relationships (96) it is seen that according to this model the Poisson’s ratio could vary in a very broad range, namely, from −1 to 1/2.

3.6 Models considering volumetric relaxation Taking the volumetric relaxation into account results in more cumbersome calculã , which are defined, respectively, by tions [14, 24, 25, 27]. Thus, if operators Ẽ and μ ̃−1 utilizing the (61) and (67), are known, then first it is necessary to find operator K relationship ̃ ̃ 1 −1 1 −1 ̃J = Ẽ −1 = 3K + μ ̃ , ̃ + K = μ ̃μ 3 9 ̃ 9K

(97)

γ γ ̃−1 = 9Ẽ −1 − 3μ ̃ −1 = a1 ∋∗γ (τσE ) − a2 ∋∗γ (τσμ ) , K

(98)

whence it follows that

where a1 = 9E0−1 and a2 = 3μ−1 0 . Now we could calculate the ratio of operators considering equations (61) and (98), i. e., Ẽ γ γ γ = E0 [1 + (τσE ) Dγ ][a1 ∋∗γ (τσE ) − a2 ∋∗γ (τσμ ) ], ̃ K or γ γ τE τE Ẽ γ = E0 [a1 − a2 ( σμ ) ] − E0 a2 [1 − ( σμ ) ]∋∗γ (τσμ ) . ̃ τσ τσ K

(99)

Computing operator ν̃ via the formula Ẽ = 3(1 − 2ν̃) ̃ K

(100)

and considering (99) yields ν̃ =

γ

γ

τE τE E 1 E0 γ − [a1 − a2 ( σμ ) ] + 0 a2 [1 − ( σμ ) ]∋∗γ (τσμ ) . 2 6 6 τσ τσ

(101)

Fractional calculus models in dynamic problems of viscoelasticity | 155

Considering the product ν(t) = ν̃ ⋅ H(t), we find γ

γ

τE τE E 1 E0 − [a1 − a2 ( σμ ) ] = −1 + 0 ( σμ ) , 2 6 2μ0 τσ τσ E0 1 E0 = ν0 . (a − a1 ) = −1 + ν(∞) = + 2 6 2 2μ0

ν(0) =

(102) (103)

Reference to relationship (102) shows that the low limiting magnitude of Poisson’s ratio ν(0) depends on the ratio of two retardation times and fractional parameter γ, as distinct from the Kelvin–Voigt model without volumetric relaxation for which ν(0) = −1 according to (77). τE

However, note that when ( τσμ )γ → 0, from (102) it follows that ν(0) → −1. It means σ that the model under consideration could be used for modeling the features of visτE

coelastic auxetic materials until ( τσμ )γ → τσE μ τσ

γ

σ

2μ0 , E0

2μ0 E0

resulting in ν(0) → 0. For magnitudes

ν(0) ≥ 0, which corresponds to traditional viscoelastic materials. ̃ (86) and The standard linear solid model with two pre-assigned Lame operators μ ̃ for which we have λ,

( ) ≥

λ γ γ

1 + (τσ ) D , λ̃ = λ0 1 + (τελ )γ Dγ

(104)

has been studied by Rossikhin et al. [27]. In this case, the bulk relaxation results in the viscoelastic Poisson’s operator ν̃, which could be computed via formula ν̃ =

λ̃ γ γ = ν∞ [1 + M1 ∋∗γ (T1 ) + M2 ∋∗γ (T2 )], ̃ ̃) 2(λ + μ

(105)

where Mi and Ti (i = 1, 2) are the material’s parameters, whence it follows that the viscoelastic Poisson’s ratio varies from ν(t)|t→0 = ν∞ to ν(t)|t→∞ = ν0 . It is also very important to note that if there exists some “hypothetical” viscoelasμ tic material with retardation times equal in magnitude, τσE = τσ , then from (102) and (103) it is seen that such a material should possess the time-independent Poisson’s ratio ν̃ = ν = const. As a result, all existing viscoelastic operators are proportional to each other, which, generally speaking, does not apply to all real viscoelastic materials, and therefore the utilization of such a so-called viscoelastic material in dynamic problems of mechanics of solids is inappropriate from the point of view of mechanics. The critical review of the role of the Poisson’s ratio in linear viscoelasticity is presented by Tschoegl et al. [32] and Hilton [8, 9], wherein it has been shown that “no dynamic effects and no body forces, as well as no moving boundaries, i. e., no penetration or ablation problems, can be included” in the consideration adopting the assumption of time-independent Poisson’s ratios. In Table 1, the limiting values of viscoelastic Poisson’s ratios are presented for different fractional derivative models, summarizing the above discussion.

156 | Y. Rossikhin and M. Shitikova Table 1: Limiting values for the viscoelastic Poisson’s ratio ν(t) = ν̃ ⋅ H(t). Fractional derivative model

ν(t)|t→0

ν(t)|t→∞

−∞

ν0

−1

1 2

−1

ν0

ν∞

1 2

ν∞

ν0

(1) Kelvin–Voigt model Ẽ = E0 [1 + (τσE )γ Dγ ], K̃ = K0 = const E0 (τ E )γ Dγ 2K0 σ

ν̃ = ν0 −

(2) Scott Blair model ̃ = μτ γ Dγ , K̃ = K = const μ ν̃ = −1 + 23 ∋∗γ (T γ ) (3) Kelvin–Voigt model ̃ = μ0 [1 + (τσμ )γ Dγ ], K̃ = K0 = const μ ν̃ = −1 +

E0 ∗ μ γ ∋ (τ ) 2μ0 γ σ

(4) Maxwell model ̃ = μ∞ [1 − ∋∗γ (τεγ )], K̃ = K∞ = const μ ν̃ = ν∞ +

E∞ ∗ γ ∋ (τ ) 6K∞ γ ε

(5) Standard linear solid model γ γ

̃ = μ0 1+τσγ Dγ , K̃ = K∞ = const μ ν̃ =

1+τε D E ∞ Mσ ν∞ + 2μ ∞

γ

∋∗γ (τσ )

(6) Kelvin–Voigt model with bulk relaxation ̃ = μ0 [1 + (τσμ )γ Dγ ] Ẽ = E0 [1 + (τσE )γ Dγ ], μ ν̃ = +

1 2

−

E0 [a1 6

E0 a [1 6 2

−(

− a2 (

τσE γ μ) ] τσ

τσE γ ∗ μ γ μ ) ]∋γ (τσ ) τσ

−1 +

(7) Standard linear solid model with bulk

E E0 τσ γ ( ) 2μ0 τσμ

ν0

relaxation μ γ γ

λ γ γ

̃ = λ 1+(τσ ) D ̃ = μ0 1+(τσμ )γ Dγ , λ μ 0 λ γ γ

1+(τε ) D γ γ M1 ∋∗γ (T1 ) + M2 ∋∗γ (T2 )]

1+(τε ) D

ν̃ = ν∞ [1 +

ν∞

ν0

4 Conclusion In the present chapter, viscoelastic operators have been constructed for the fractional derivative Kelvin–Voigt, Maxwell, and standard linear solid models with and without volume relaxation involving time-dependent Poisson’s operators, since these models are frequently used in engineering applications. The suitability of these models for real advanced materials including auxetics is evaluated.

Fractional calculus models in dynamic problems of viscoelasticity | 157

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Yury Rossikhin and Marina Shitikova

Fractional calculus in structural mechanics Abstract: In the present chapter, the viscoelastic fractional-order operators suitable for solving dynamics problems of mechanics of solids and structural mechanics are analyzed by the examples of linear and non-linear fractional oscillators, beams, and plates, as well as impact problems, which are frequently used in engineering applications. It has been shown that nowadays fractional calculus has entered the mainstream engineering analysis and is widely applied to structural dynamics problems. The place and role of fractional calculus in engineering problems and practice have been revealed. Keywords: Viscoelasticity, Riemann–Liouville fractional derivative, Rabotnov fractional exponential function, viscoelastic fractional operators, time-dependent Poisson’s operator MSC 2010: 65C05, 62M20, 93E11, 62F15, 86A22

1 Some historical remarks Investigation of properties of the fractional operators with the aim of applying them to the problems of mechanics of solids was initiated in the 1940s. A chronological list of the main contributions made prior to 1980 by Western and Russian researchers in the field of fractional calculus applications in linear viscoelasticity is presented by Rossikhin in his retrospective article [36], which shows to the present generation of researchers applying fractional calculus in many branches of science that the work of these pioneers was primarily responsible for the development of the theory of fractional calculus viscoelasticity. In Table 1, the simplest fractional calculus equations modeling the viscoelastic features of materials using the two approaches, i. e., the Boltzmann–Volterra relationships with weakly singular kernels and fractional derivatives or fractional integrals, as well as other types of fractional operators, are presented in the first column, while the Russian and Western authors who proposed them for the first time (to our knowledge) in problems of viscoelasticity are cited in the second and third columns, respectively. The last column shows the papers wherein these Acknowledgement: This work is partially supported by the Russian Foundation for Basic Research, Project No. 17-01-00490 and partially by the Ministry of Education and Science of the Russian Federation (including the current Project No. 9.5138.2017/8.9). Yury Rossikhin, Marina Shitikova, Research Center on Dynamics of Solids and Structures, Voronezh State Technical University, 20-letiya Oktyabrya Str. 84, Voronezh 394006, Russian Federation, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110571905-009

−γ

τε Iγ

−γ

1+τε Iγ

]ε

ε = J0 ∫0 ∋γ (−t 󸀠 /τσ )σ(t − t 󸀠 )dt 󸀠 (b) via fractional derivatives γ σ = E0 (ε + τσ Dγ ε)

t

Kelvin–Voigt model (a) via Boltzmann–Volterra relationships

σ = E∞ [1 − νε

(b) via fractional derivatives γ γ σ + τε Dγ σ = E0 (ε + τσ Dγ ε) (b) via fractional operators

ε = J∞ [σ + νσ ∫0 ∋γ (−t 󸀠 /τσ )σ(t − t 󸀠 )dt 󸀠 ]

t

σ = E∞ [ε − νε ∫0 ∋γ (−t 󸀠 /τε )ε(t − t 󸀠 )dt 󸀠 ]

t

standard linear solid model (a) via Boltzmann–Volterra relationships

σ(t) =

γ E∞ τε I1−γ dε(t)

fractional Newtonian model γ σ(t) = E∞ τε Dγ ε(t)

The simplest fractional calculus models

Shermergor [69]

Shermergor [69]

Rabotnov [34]

Meshkov [27]

Rabotnov [33]

Gerasimov [16]

Russian researchers

Caputo [9] Smit and de Vries [71]

Caputo and Mainardi [12]

Gross [20]

Bland [6]

Scott Blair [68]

Western researchers

Caputo [9, 10] Bagley and Torvik [3]

Caputo and Mainardi [12, 13]

Meshkov and Rossikhin [29] Zelenev et al. [76] Meshkov et al. [30] Gonsovskii et al. [17] Gonsovskii and Rossikhin [19, 18]

Rozovskii and Sinaiskii [65]

Caputo [11]

Applications carried out prior to 1980

Table 1: A historical overview of Russian and Western contributors to the use of fractional calculus in mechanics of solids, from the 1940s to the 1970s [36].

160 | Y. Rossikhin and M. Shitikova

t

󸀠

σ = E∞ [ε − νε ∫0

󸀠 t t γ−1 e−t /τε γ τε Γ(γ)

(b) as a relaxation kernel

σ = E∞ [ε −

ε = J∞ [σ + νσ ∫0

ε(t − t 󸀠 )dt 󸀠 ]

t t γ−1 e−t /τσ σ(t − t 󸀠 )dt 󸀠 ] γ τσ Γ(γ) 󸀠 t ∫0 e−t /τσ ∋γ (−νσ t 󸀠 /τσ )ε(t − t 󸀠 )dt 󸀠 ]

fractional operator with Rzhanitsyn kernel (a) as a creep kernel d −γ ) ]σ ε = J∞ [1 + νσ (1 + τσ dt

σ = E∞ [ε − ∫0 ∑nj=1 ej ∋γ (−t 󸀠 /τε j )ε(t − t 󸀠 )dt 󸀠 ]

Meshkov and Rossikhin [30]

Vulfson [74]

Rzhanitsyn [66]

Meshkov [27] Slonimsky [70]

Rabotnov [34]

Shermergor [69]

(c) via fractional integral −γ ε = J∞ (σ + τε Iγ σ)

generalized standard linear solid model

Meshkov [27]

Shermergor [69]

Russian researchers

(b) via fractional derivatives σ + τε1/2 D1/2 σ = ηε̇ γ γ σ + τ ε Dγ σ = E ∞ τ ε Dγ ε

σ = E∞ [ε − ∫0 ∋γ (−t 󸀠 /τε )ε(t − t 󸀠 )dt 󸀠 ]

t

Maxwell model (a) via Boltzmann–Volterra relationships

The simplest fractional calculus models

Table 1: (continued).

Gemant [15] Caputo and Mainardi [13]

Western researchers

Belov and Bogdanovich [5]

Meshkov and Rossikhin [30]

Buchen and Mainardi [8]

Zelenev et al. [75]

Applications carried out prior to 1980

Fractional calculus in structural mechanics | 161

162 | Y. Rossikhin and M. Shitikova models were implemented for the first time for solving different dynamic problems of mechanics of solids and geophysics. The enumeration of the models started from the fractional Newtonian model suggested by Scott Blair [68] and Gerasimov [16] using the second and first approaches, respectively. This simplest fractional calculus element was further named by Koeller [23] as the spring-pot with the memory parameter γ, because it has the property that its constitutive equation has continuity from the ideal solid state at γ = 0, to the ideal fluid state at γ = 1. This model was used by Caputo [11] to study vibrations of an infinite viscoelastic layer. The fractional calculus standard linear solid model proposed by Gross [20] and Rabotnov [33] was employed with great success in the 1970s by Russian scholars for solving a rich variety of dynamic engineering problems [36]. Thus, in 1966, Rozovskii and Sinaiskii [65] were the first to study the impulse response of an oscillator representing the order of the fractional exponent as the proper fraction, γ = q/n, where q and n are integers, that results in the rationalization of the corresponding characteristic equation. This procedure is now widely used by the authors dealing with numerical algorithms in treating equations with fractional operators. The Green functions for the fractional standard linear solid and Maxwell oscillators were obtained in Zelenev et al. [75] and Zelenev et al. [76], respectively. As for the wave propagation in hereditarily elastic media, harmonic bulk waves were investigated by Meshkov and Rossikhin [29], and the impulse load propagation in a rod made of the fractional standard linear solid material was studied by Gonsovskii and Rossikhin [19]. The impact of a hereditarily elastic rod with the Rabotnov kernel of heredity against a rigid barrier was studied by Gonsovskii et al. in 1972 [17]. The nearbyfront asymptotic of the stress waves in a semi-infinite rod originated from the constant load application to its free end at the initial time was obtained by Gonsovskii and Rossikhin [18]. Two years later, the asymptotic solution in the neighborhood of the step pulse onset in a similar problem for a viscoelastic rod made of the fractional Maxwell material was presented by Buchen and Mainardi [8], using the second approach for γ = 1/2. Rabotnov [34] was a groundbreaker in the generalization of the fractional calculus standard solid model, suggesting in 1966 the rheological models which describe the dissipative processes with several relaxation (retardation) times based on the fractional exponential function suggested by him in [33]. He further used such models in [35] for an approximation of the relaxation and creep functions determined experimentally for several polymeric materials. The presence of a large variety of rheological parameters allows one to adequately describe the behavior of the advanced polymeric materials possessing the complex relaxation and creep functions involving several plateau and several transition zones. Rzhanitsyn [66] proposed in 1949 another weakly singular kernel of heredity as a creep kernel, which was written in the form of a fractional operator different from the fractional derivative or fractional integral. Its resolvent kernel of relaxation was

Fractional calculus in structural mechanics | 163

found by Vulfson in 1960 [74]. In 1967, Meshkov [27] suggested the equivalent form for the Rzhanitsyn model, while Slonimsky [70] employed it for describing relaxation processes in polymers with a chain structure of macro-molecules. Four years later, Meshkov and Rossikhin [30] suggested to use the Rzhanitsyn kernel as the relaxation kernel in the Boltzmann–Volterra equation and utilized this model to study the impulse response of a viscoelastic oscillator. Belov and Bogdanovich [5] used the model proposed in [30] for investigating the propagation of a load impulse in a semi-infinite rod and constructed its asymptotic solution. The fractional derivative Kelvin–Voigt and standard linear solid models were first proposed by Shermergor [69] and Meshkov [27] in 1966 and 1967, respectively, and then independently but at a somewhat later time by Caputo [9] and Caputo and Mainardi [12], respectively. The early applications of these models were made by Caputo [9] in 1967 and Caputo and Mainardi [12, 13] in 1971 for solving the problems dealing with geophysics. In 1974, the fractional derivative Kelvin–Voigt model was utilized by Caputo [10] to study vibrations of an infinite viscoelastic layer. In the late 1970s, an investigation on fractional derivative models and their application to the problems of structural dynamics was initiated in the USA by Torvik and Bagley. Their first paper in the field [3], dated 1979, was devoted to modeling an elastomer damper as a fractional derivative Kelvin–Voigt oscillator. During the next decade, these authors contributed significantly to incorporate the fractional derivative standard linear solid model into the numerical procedures for investigating viscoelastically damped structures [4, 73, 72]. Thus, from the papers cited in the forth column of Table 1, it is clearly evident what a tremendous work was carried out in the late 1960s and 1970s by Russian and Italian researchers in the application of fractional calculus viscoelastic models (formulated earlier or in the same period using the two approaches as shown in the first three columns of Table 1) for solving dynamic problems in the mechanics of solids and geophysics.

2 Vibrations of oscillators on the basis of fractional operator viscoelastic models The necessity of studying fractional oscillators is motivated by two reasons: first, engineers often use one-degree-of-freedom models as a first approximation or as a benchmark before proceeding to more intricate models or multi-degree-of-freedom structural systems, for example, as the simplest model of a vibration-isolation system [24, 26], and second, the study of vibrations of more complex structures can be reduced to vibrations of a set of oscillators [1, 39, 40, 45]. Assume that viscoelastic features of operators are described by the Boltzmann– Volterra relationships with the relaxation kernel Kε (t) and retardation kernel Kσ (t).

164 | Y. Rossikhin and M. Shitikova Then the equations of motion of such an oscillator could be written in two different but equivalent forms [37], i. e., ẍ +

ω2∞ [x

t

− νϵ ∫ Kϵ (t − t 󸀠 )x(t 󸀠 )dt 󸀠 ] = Fδ(t),

(1)

0

t

̈ 󸀠 )dt 󸀠 = F[δ(t) + νσ Kσ (t)], ẍ + ω2∞ x + νσ ∫ Kσ (t − t 󸀠 )x(t

(2)

0

where x is the coordinate, F is the amplitude of force impulse per unit mass, νε = −1 △EE∞ , △E = E∞ − E0 is the defect of the modulus, i. e., the value characterizing the decrease in the elastic modulus from its non-relaxed value E∞ to its relaxed value E0 , −1 , △J = J0 − J∞ , J∞ and J0 are the non-relaxed and relaxed magnitudes of νσ = △JJ∞ compliance, respectively, ω∞ is the frequency of elastic vibrations corresponding to the non-relaxed magnitude of the elastic modulus, and overdots denote time derivatives. Applying the Laplace transformation to equations (1) and (2) yields ̄ x(p) =

p2

+

F[1 + νσ K̄ σ (p)] F = . 2 − νϵ K̄ ϵ (p)] ω∞ + p2 [1 + νσ K̄ σ (p)]

ω2∞ [1

(3)

The solution of (3) in the time domain according to the inversion Mellin–Fourier formula has the form c+i∞

x(t) =

1 pt ̄ dp. ∫ x(p)e 2πi

(4)

c−i∞

̄ considered here are multi-valued functions with the branch points Functions x(p) p = 0 and p = −∞ or p = −s∗ , s∗ ≥ 0, and p = −∞, that is, the inversion should be carried out on the first sheet of a Riemannian surface with a cut along the real negative semi-axis from 0 to −∞ or from −s∗ to −∞. Figure 1 shows the closed contour of integration for the function with the branch points p = 0 and p = −∞. If a function possesses the branch points p = −s∗ and p = −∞, then the center of a small circumference should locate at the point p = −s∗ . Due to the Jordan lemma, the curvilinear integrals taken along the arcs cR tend to zero at R → ∞. For weakly singular kernels, ̄ the integral taken along cρ also tends to zero when ρ → 0. Besides, the function x(p) possesses the ordinary poles at the same magnitudes of p which make the denominator vanish in equation (3), i. e., they are the roots of the characteristic equations p2 + ω2∞ [1 − νϵ K̄ ϵ (p)] = 0,

ω2∞

2

+ p [1 + νσ K̄ σ (p)] = 0.

(5) (6)

Fractional calculus in structural mechanics | 165

Figure 1: Contour of integration.

Defining the variable p as p = reiψ and introducing notations ρα = √ℜ2α [ ] + ℑα2 [ ],

ℑα [ ] (α = 1, 2), ℜα [ ] ℑ1 [1 − νε K̄ ε (p)] = ℑ1 [ ], ℑ2 [1 + νσ K̄ σ (p)] = ℑ2 [ ],

tan χα =

ℜ1 [1 − νε K̄ ε (p)] = ℜ1 [ ], ℜ2 [1 + νσ K̄ σ (p)] = ℜ2 [ ], we obtain

i(2ψ−χ1 ) + ω2∞ = 0, r 2 ρ−1 1 e

i(2ψ+χ2 ) + ω2∞ = 0. r 2 ρ−1 2 e

(7) (8)

Separating the real and imaginary parts in equations (7) and (8), we find r 2 ρ−1 cos (2ψ − χ1 ) + ω2∞ = 0, { 2 1−1 r ρ1 sin (2ψ − χ1 ) = 0,

(9)

r 2 ρ2 cos (2ψ + χ2 ) + ω2∞ = 0, {2 r ρ2 sin (2ψ + χ2 ) = 0.

(10)

From the sets of equations (7) and (8), we find 2ψ − χ1 = ±π, { 2 −1 r ρ1 = ω2∞ ,

(11)

2ψ + χ2 = ±π, {2 r ρ2 = ω2∞ .

(12)

From relationships (11) and (12), we find the values of r and ψ, which are, respectively, the modulus and argument of the root of characteristic equations (5) and (6).

166 | Y. Rossikhin and M. Shitikova Using the Jordan lemma and the main theorem of the theory of residues, the solution to equations (11) and (12) may be written as x(t) = xdrift (t) + xvibr (t) =

∞

1 ̄ k )epk t ], ̄ −iπ ) − x(se ̄ iπ )]e−st ds + ∑ res[x(p ∫ [x(se 2πi k

(13)

0

x(t) =

∞

1 ̄ k )epk t ], ̄ −iπ ) − x(se ̄ iπ )]H(s − s∗ )e−st ds + ∑ res[x(p ∫ [x(se 2πi k

(14)

0

where H(s − s∗ ) is the Heaviside function and the summation is taken over all isolated singular points (poles). Since characteristic equations possess, as a rule, two complex conjugate roots p1,2 = −α ± iω = re±iψ ,

(15)

relationships (13) and (14) take the form x(t) = A0 (t) + A exp(−αt) sin(ωt − φ),

(16)

where ∞

A0 (t) = ∫ τ−1 B(τ)et/τ dτ,

(17)

0

and B(τ) is the function of distribution of the relaxation parameters (retardation parameters) of the dynamic system. From equation (16) it is seen that the relationship describing vibrations of an oscillator with the natural frequency ω and the damping factor α possesses two terms, one of which describes the drift of the equilibrium position and is represented by the integral involving the distribution function of dynamic and rheological parameters, while the other term is the product of two time-dependent functions, exponent and sine, and it describes damped vibrations around the drifting equilibrium position. In doing so, the drift is defined by the function A0 (t). The first term xdrift (t) is governed by an improper integral taken along two sides of the cut along the negative real semiaxis of the complex plane (see Figure 1), while the second term xvibr (t) is determined by two complex conjugate roots of the characteristic equation, which locate in the left half-plane of the complex plane. The original method for solving a characteristic equation with fractional powers without its rationalization has been suggested in [28, 76], when the roots and model’s parameters (relaxation or retardation times) are found at a time. It has been established that the characteristic equation lacks real negative roots and possesses only two complex conjugate roots. The behavior of the characteristic equation roots in the complex plane depends on the type of relaxation or creep kernel involved in the model

Fractional calculus in structural mechanics | 167

of an oscillator under consideration. The interested reader is referred to [37, 40, 45] for a fully comprehensive description of the mathematical formulation. Thus, in order to obtain the final solution, it is necessary to specify the type of the relaxation or creep kernel. We consider a few such examples. If as an example the Rabotnov model [54] for the stress (σ)-strain (ε) relationship with the relaxation time τε , i. e., t

σ = E∞ [ε − νε ∫ ∋γ (−s/τε )ε(t − s)ds],

(18)

0

is utilized as a relaxation kernel Kε (t), where ∋γ (−t/τi ) is the Rabotnov fractional exponential function [33] ∋γ (−t/τε ) =

t γ−1 ∞ (−1)n (t/τε )γn , γ ∑ τε n=0 Γ[γ(n + 1)]

(19)

then the solution (13) takes the form of equation (16) [37], where 1/2

A = 2F[(a2 + b2 ) (τϵ−2γ + r 2γ + 2τϵ−γ r γ cos γψ)] , −1

tan φ = −

τϵ cos β + r γ cos(β − γψ) −γ

−γ τϵ

sin β +

rγ

sin(β − γψ)

,

β=

b , a

r 2 = ω2 + α 2 ,

(20) ω tan ψ = − , α

a = (2 + γ)r 1+γ cos(1 + γ)ψ + 2τϵ−γ r cos ψ + γω2∞ r γ−1 cos(γ − 1)ψ, b = (2 + γ)r 1+γ sin(1 + γ)ψ + 2τϵ−γ r sin ψ + γω2∞ r γ−1 sin(γ − 1)ψ, and B(τ, τϵ−γ ) =

νϵ ω2∞ F[θ∞ (τ)]−1 [θ0 (τ)]−1 τ3 sin γπ , π θ∞ (τ)[θ0 (τ)]−1 τ−γ τϵ−γ + θ0 (τ)[θ∞ (τ)]−1 τγ τϵγ + 2 cos πγ θ∞ (τ) = τ2 ω2∞ + 1,

(21)

θ0 (τ) = τ2 ω20 + 1,

and ω0 is the frequency of elastic vibrations corresponding to the relaxed magnitude of the elastic modulus. The behavior of the roots of the characteristic equation (5) as function of the pa−γ rameter τϵ is shown in Figures 2a–d for ω∞ = 1 and four values ξ = 0, 1/50, 1/9, and 1/6, respectively, where figures near curves denote the magnitudes of the value γ and ξ = E0 /E∞ = 1 − νϵ . −γ It is seen that the τϵ -dependence of the two complex conjugate roots p1,2 = −α±iω −γ at γ ≠ 1 leave the points ±i and converge in the points ±iξ 1/2 when τϵ changes from 0 to ∞; in doing so it does not meet the real negative semi-axes and remains inside the −γ curves for the τϵ -dependencies of three roots of the characteristic equation with γ = 1 (the ordinary standard linear solid model). The behavior of the two roots of three at

168 | Y. Rossikhin and M. Shitikova

Figure 2: Behavior of the complex conjugate roots p1,2 = −α ± iω for an oscillator based on the Rabotnov model. (a) ξ = 0; (b) ξ = 1/50; (c) ξ = 1/9; and (d) ξ = 1/6.

Fractional calculus in structural mechanics | 169

γ = 1 (the third root is the real root all the time and changes from 0 to ∞ as τϵ varies from 0 to ∞) essentially depends on the magnitude of the value ξ . Thus, at the values ξ taken from the interval (0, 1/9) the two complex conjugate roots first become real as −γ −γ τϵ changes from 0 to ∞. But as τϵ further increases, they again become complex −γ conjugate roots, i. e., the domain of aperiodicity (the values of τϵ wherein ω = 0) has finite dimensions contracting with increase in ξ (Figures 2a, b). At ξ = 0 the ordinary Maxwell model with a semi-infinite domain of aperiodicity is obtained (Figure 2a) [37], but at ξ = 1/9 the domain of aperiodicity degenerates into a point (Figure 2c). When ξ > 1/9, the domain of aperiodicity completely disappears (Figure 2d). In other words, at γ ≠ 1 the behavior character of the roots of the characteristic equation for the Rabotnov model is governed by the magnitudes of the value ξ , which may be considered as the deficiency of the elastic modulus. The procedure described has been used for the analysis of fractionally damped oscillators based on different fractional operators with one or more fractional parameters and one or more relaxation/retardation times: the Kelvin–Voigt model, Maxwell model, Rzhanitsyn model, and generalized Rabotnov model [37, 41, 43, 45, 46, 52]. −γ

3 Linear vibrations of beams and plates Usually in the papers related to the dynamic response of viscoelastic bodies such as beams, plates, and membranes, the Kelvin–Voigt model with fractional derivatives is utilized [2, 21, 31, 77]. As this takes place, it is supposed that the Poisson’s ratio is time-independent during the process of deformation and as a pre-assigned operator the chosen Young’s operator Ẽ is defined by γ Ẽ = E0 [1 + (τσE ) Dγ ],

(22)

where τσE is the retardation, or creep, time, and Dγ is the Riemann–Liouville fractional derivative. However, it has been recently shown for viscoelastic beams [55] and plates [61] that when operator Ẽ is defined by equation (22) and the Poisson’s operator ν̃ is considered as the time-independent value, this case coincides with the case of the dynamic behavior of elastic bodies in a viscoelastic medium, damping features of which are defined by the fractional derivative Kelvin–Voigt model.

3.1 Vibrations of the Bernoulli–Euler beam Let us consider a freely supported viscoelastic Bernoulli–Euler beam, the viscoelastic features of which are described by a fractional derivative Kelvin–Voigt model (22). The

170 | Y. Rossikhin and M. Shitikova equations of motion of such a beam have the form 4 ̃ 𝜕 w(x, t) + ϱAw(x, ̈ t) = q(x, t), EI 𝜕x 4

(23)

where w(x, t) is the displacement of the beam at the contact point, I is the moment of inertia of the beam’s cross-section, A is the beam’s cross-sectional area, ϱ is its density, x is the longitudinal coordinate, an overdot denotes partial time derivative, and q(x, t) is the load. Expanding the displacement w(x, t) in terms of eigenfunctions Wn (x), we have ∞

w(x, t) = ∑ Wn (x)Tn (t),

(24)

n=1

where Tn (t) are generalized displacements, Wn (x) = sin(

nπ x), L

(25)

and L is the beam’s length. Substituting (24) in equation (23) and considering the orthogonality condition for the eigenfunctions (25) on the segment from 0 to L, we are led to the infinite set of uncoupled equations T̈ n (t) + Ω2n (1 + τσγ Dγ )Tn (t) = Pn (t)

(n = 1, 2, . . .),

(26)

each of which describes force-driven vibrations of the viscoelastic oscillator, where Ω2n

4

E I nπ = 0 ( ) , ϱA L

L

Pn (t) =

∫0 q(x, t)Wn (x)dx L

ϱA ∫0 [Wn (x)]2 dx

.

Now we will show that the equation equal to equation (26) could be obtained if one considers the problem on vibrations of an elastic beam in a viscoelastic medium [55], when the equation of motion of such a beam has the form E0 I 𝜕4 w(x, t) μ γ 1 ̈ t) = + D w + w(x, q(x, t), ρA 𝜕x 4 ρA ρA

(27)

where μ is the viscosity coefficient. Substituting (24) in equation (27) and considering the orthogonality condition for the eigenfunctions (25) on the segment from 0 to L, we are led to the infinite set of uncoupled equations μ T̈ n (t) + n Dγ Tn (t) + Ω2n Tn (t) = Pn ρA where μn is the viscous coefficient of the nth mode.

(n = 1, 2, . . .),

(28)

Fractional calculus in structural mechanics | 171

Considering the Rayleigh hypothesis about the proportionality between the elastic and viscous matrices, i. e., μn = Ω2n τσγ , (29) ρA γ

where τσ is the coefficient of proportionality, equation (28) is reduced to equation (26). Thus, our assertion has been proven.

3.2 Vibrations of the Kirchhoff–Love plate In the third volume of the three-volume handbook “Strength. Stability. Vibrations”, edited by Birger and Panovko [7], there is a subsection devoted to the consideration of external and internal friction during vibrations of plates, wherein it is written that the simplest model taking the influence of external friction into account is based on the hypothesis that the intensity of the external friction force is proportional to the velocity of the plate’s deflection at the given point (in our case the conventional time derivative is substituted by the fractional time derivative). The corresponding differential equation of vibrations of a plate with constant thickness h has the form ̈ t) = q(x, y, t), D󳶚2 w(x, y, t) + μDγ w(x, y, t) + ρhw(x,

(30)

where D = Eh3 [12(1 − ν2 )]−1 is the cylindrical rigidity, μ is the viscosity coefficient, and 󳶚2 = (𝜕2 /𝜕x2 + 𝜕2 /𝜕y2 ). From a large amount of models reflecting the influence of internal friction the simplest and probably the most wide-spread is the Kelvin–Voigt model. The differential equation of vibrations of a plate made of such a material has the form ̈ t) = q(x, y, t), D0 󳶚2 [w(x, y, t) + τσγ Dγ w(x, y, t)] + ρhw(x,

(31)

where D0 = E0 h3 [12(1 − ν2 )]−1 . ̃ entering in the cylindrical rigidity, takes the In other words, the Young operator E, form of (22), while the Poisson’s ratio remains a constant value. Expanding the displacement w(x, y, t) in terms of eigenfunctions ∞ ∞

w(x, y, t) = ∑ ∑ Wmn (x, y)xmn (t), n=1 m=1

(32)

where xmn (t) are generalized displacements, while Wmn (x, y) = sin(

nπy mπx )( ), a b

(33)

substituting (32) in equation (31) and considering the orthogonality condition of functions (33) on the segments 0 ≤ a and 0 ≤ y ≤ b, we have ẍmn (t) + Ω2mn (1 + τσγ Dγ )xmn (t) = Pmn (t),

(34)

172 | Y. Rossikhin and M. Shitikova where Ω2mn

2

2

D m n = 0 [( ) + ( ) ], ϱh a b

b

a

Pn (t) =

∫0 ∫0 q(x, y, t)Wmn (x, y)dxdy a

b

ϱh ∫0 ∫0 [Wmn (x, y)]2 dxdy

.

Now consider vibrations of an elastic plate in a viscous medium with fractional viscosity. Then, substituting (32) in equation (30) preliminary divided by ρh and considering the orthogonality condition of functions (33) on the segments 0 ≤ a and 0 ≤ y ≤ b, we have ẍmn (t) +

μmn γ D xmn (t) + Ω2mn xmn (t) = Pmn (t), ρh

(35)

where μmn is the viscosity coefficient of the mnth harmonic. Considering the Rayleigh hypothesis about proportionality of elastic and viscous matrices, i. e., μmn = Ω2mn τσγ , ρh

(36)

γ

where τσ is the coefficient of proportionality, equation (35) is reduced to equation (34). Thus, our assertion turns out to be valid, while the assumption in [7] is questionable. This conclusion is supported by another fact. When solving boundary value problems of viscoelasticity connected with the dynamic response of plates and shells, it is ̃ proportional to the cylindrical rigidity of the plate. In needed to find the operator d the case under consideration, it has been calculated in [57] that ̃= d

γ γ τσE τσE 1 E0 1 E0 Ẽ ( ( ) ) + = 2 1 − ν0 τσν 1 − 2ν̃2 2 1 + ν0 τεν γ

+

τE 1 E0 γ [( σν ) + 1]∋∗γ ((−τεν ) ) 2 1 + ν0 τε γ

+

τE 1 E0 γ [1 − ( σν ) ]∋∗γ ((τσν ) ), 2 1 − ν0 τσ

(37)

where ∋∗γ (τσγ ) =

1 γ 1 + τσ Dγ

(38)

is the dimensionless Rabotnov fractional operator [49, 51, 52] and γ

(τεν ) =

E0 (τσE )γ , 6K0 (1 + ν0 )

γ

(τσν ) =

E0 (τσE )γ . 6K0 (1 − ν0 )

Fractional calculus in structural mechanics | 173

̃ involves the operator ∋∗ ((−τν )γ ), From equation (37) it is seen that the operator d γ ε which as a memory kernel has an increasing with time function ∋γ (t/τσ ) =

t γ−1 ∞ (t/τσ )γn , γ ∑ τσ n=0 Γ[γ(n + 1)]

(39)

which is in contradiction with the principle of fading memory.

4 Problems of shock interaction of viscoelastic bodies Problems of shock interaction of viscoelastic bodies belong to the most intricate problems of that kind. Below we will consider two approaches for modeling the contact force mainly in the problems of shock interaction of thin viscoelastic bodies with rigid, elastic, and viscoelastic bodies of finite dimensions. Impactors are usually modeled by a sphere or a rod. In 1973, Rozovskii [64] was the first to consider the problem of impact of a perfectly rigid flat-ended stamp on a large viscoelastic plate, the viscoelastic features of which were described by a Rabotnov fractional operator, and he was also the first to study the case when a spring with a linear characteristic is inserted between the impacting stamp and the plate. During the last decade more than 20 papers have been published dealing with the shock interaction of thin bodies with bodies of finite dimensions. Some of them applied the linear approach for modeling the contact force via the fractional derivative buffers [32, 44, 45, 47, 14, 52, 50], and some applied the non-linear approach via the generalized Hertzian contact law by introducing the fractional operators for cylindrical rigidity and time-dependent Poisson’s ratios [21, 49, 55, 61, 58, 60].

4.1 Linear approach via a fractional derivative spring-dashpot buffer Within the linear contact law, a viscoelastic buffer with a fractional derivative constitutive equation either via the Maxwell model [44] or via the standard linear solid model [47] could be utilized for the analysis of the impact response of an elastic plate. Thus, an impactor with mass m falls down with the initial velocity V0 against a circular elastic Uflyand–Mindlin plate (Figure 3), the equations of motion of which have the form [47] 𝜕Qr 1 ̇ + Qr = ρhW, 𝜕r r

𝜕W Q̇ r = Kμh( − Br ), 𝜕r

(40)

174 | Y. Rossikhin and M. Shitikova 𝜕Mr 1 ρh3 ̇ B, + Mr + Qr = 𝜕r r 12 r

𝜕B B Ṁ r = D( r + ν r ), 𝜕r r

(41)

where Qr is the transverse force, Mr is the bending moment, W is the velocity of the plate’s deflection, Br is the velocity of the rotation angle in the r-coordinate direction, K = π 2 /12 is the shear coefficient, and D = Eh3 /12(1 − ν2 ) is the cylindrical rigidity.

Figure 3: The scheme of the shock interaction of an impactor with an elastic plate via a viscoelastic buffer.

Impact occurs at t = 0, resulting in the generation of waves of strong discontinuity: the longitudinal-flexural wave Σ1 and the transverse shear wave Σ2 , which propagate, respectively, with the velocities G(1) and G(2) [47], given by G(1) = √

E , ρ(1 − ν2 )

(42)

Kμ . ρ

(43)

G(2) = √

Since the process of impact is of short duration, the solution from the contact domain could be constructed in terms of one-term ray expansions [47]. If the bending of the contact region, which has the radius r0 (Figure 3), is neglected, then the transverse force could be defined as Qr = −ρhG(2) W. as

(44)

Forasmuch the contact spot moves as a rigid whole, its equation could be written ρhπr02 ẅ = 2πr0 Qr |r=r0 + Fcont ,

(45)

and the contact force satisfies the equation Fcont + τεγ Dγ F(t) = E∞ [x(t) + τσγ Dγ x(t)],

(46)

where x = α − w and α and w are the displacements of the upper and lower ends of the damper, respectively.

Fractional calculus in structural mechanics | 175

Relationship (46) could be rewritten in the form [47] t

Fcont = E∞ (α − w) − ΔE ∫ ∋γ (− 0

t − t󸀠 )[α(t 󸀠 ) − w(t 󸀠 )]dt 󸀠 . τε

(47)

Adding the equation of motion of the impactor m(ẅ + α)̈ = −Fcont

(48)

and the initial conditions α|t=0 = w|t=0 = w|̇ t=0 = 0,

α|̇ t=0 = V0

(49)

to relationships (44)–(47), we obtain the closed set of two equations in terms of two unknown values: α(t) and w(t). We have ẅ = −Bẇ + Ω2∞ (α − w) − (Ω2∞ − Ω20 )(1 + τεγ Dγ ) (α − w),

(50)

−1

ẅ + α̈ = −ω2∞ (α − w) − (ω2∞ − ω20 )(1 +

−1 τεγ Dγ ) (α

− w),

(51)

where ω20 =

E0 , m

ω2∞ =

E∞ , m

Ω20 =

E0 , M

Ω2∞ =

E∞ , M

B = 2r0−1 G(2) , and M = ρπr02 h is the mass of the contact domain. Applying Laplace transformation to equations (50) and (51) and considering equation (48) in the Laplace domain yields (p + B)(τσ + pγ ) , F̄cont (p) = V0 E∞ fγ (p) −γ

fγ (p) = (p3 + Bp2 + Cω2∞ p + Bω2∞ )(τε−γ + pγ ) − τε−γ (B + pC)ω2∞ (1 −

(52) ω20 ), ω2∞

(53)

where C = 1 + 2mM −1 . To determine the contact force Fcont we will utilize the Mellin–Fourier inversion formula (4). Considering the multiple meaning of equation (52) and assuming that equation fγ (p) = 0

(54)

possesses four complex conjugate roots pj = rj e±iψj = −αj ± iωj

(j = 1, 2),

(55)

176 | Y. Rossikhin and M. Shitikova after utilizing the integration over the contour presented in Figure 1, we obtain the solution in the form of (13), which could be finally written as 2

Fcont (t) = A0 (t) + ∑ Aj exp(−αj t) sin(ωj t + φj ),

(56)

j=1

where all values entering in this equation can be found in [47]. This approach has been extended for studying the impact response of a spherical shell [14] and for modeling the collision of two viscoelastic spherical shells via the fractional derivative standard linear solid model damper [50].

4.2 Non-linear approach via generalized Hertzian contact law To illustrate the second approach for the analysis of the impact response of viscoelastic bodies, we consider the problem on a transverse impact of a viscoelastic sphere upon a viscoelastic Bernoulli–Euler beam (Figure 4), the viscoelastic features of which are governed by viscoelastic operators corresponding to the fractional derivative standard linear solid model [60].

Figure 4: The scheme of the shock interaction of an impactor with a viscoelastic Bernoulli–Euler beam.

Then the equation of motion of a viscoelastic spherical impactor of radius R has the form mẅ 2 = −P(t),

(57)

and the motion of the viscoelastic beam of length L under the action of the contact force P(t) is defined by equation (23), where q(x, t) = P(t)δ(x − L2 ), the operator Ẽ is described by Ẽ = E∞ [1 − νε ∋∗γ (τεγ )], γ

(58) γ

γ

and ∋∗γ (τε ) is the dimensionless Rabotnov fractional operator (38) with τε instead of τσ .

Fractional calculus in structural mechanics | 177

From hereafter the subindices 1 and 2 refer to the viscoelastic beam (target) and viscoelastic sphere (impactor), respectively. Equations (57) and (23) are subjected to the following initial conditions: w1 (x, 0) = 0,

ẇ 1 (x, 0) = 0,

w2 (0) = 0,

ẇ 2 (0) = V0 ,

(59)

where V0 is the initial velocity of the impactor at the moment of impact. Integrating equation (57) two times with respect to time, we obtain t

1 w2 (t) = − ∫ P(t 󸀠 )(t − t 󸀠 )dt 󸀠 + V0 t, m

(60)

0

while the deflection of the beam w1 (t) is governed by relationship (24). Substituting the solution for a simply supported Bernoulli–Euler beam w1 (t) (24) in equation (23) and utilizing the orthogonality condition for the eigenfunctions x) on the segment from 0 to L, as a result we obtain the following infinite set of sin( nπ L uncoupled equations: −1 ̃ T̈ n (t) + Ω2∞n E∞ ETn (t) = Fn P(t)

(n = 1, 2, . . .),

(61)

each of which describes force-driven vibrations of a viscoelastic oscillator, where Ω2∞n =

4

E∞ I nπ ( ) , ϱA L

Fn =

nπ 2 sin . ϱAL 2

Knowing the Green function for the fractional derivative standard linear solid model oscillator [37], the solution for w1 (x, t) could be written in the form ∞

t

w1 (x, t) = ∑ sin( n=1

tion

nπ x) ∫ Gn (t − t 󸀠 )P(t 󸀠 )dt 󸀠 . L

(62)

0

Note that using the limiting theorem for the Laplace transform of the Green funclim pḠ n (p) = Gn (0) = 0,

p→∞

we see that at t = 0 the Green function is zero, i. e., Gn (t) for small times t is proportional to t. The contact force could be determined via the modified Hertz law ̃ 3/2 , P(t) = ky

(63)

where y is the value characterizing the relative approach of the sphere and beam, i. e., indentation of the beam by the sphere, L y(t) = w2 (t) − w1 ( , t), 2

(64)

178 | Y. Rossikhin and M. Shitikova and k̃ is the operator of rigidity involving the geometry of the impactor and the viscoelastic features of the impactor and the target, 4 k̃ = √RE ∗ , 3

(65)

which are governed via the Volterra correspondence principle by the operator E ∗ , so we have 1 − ν̃22 1 − ν̃12 1 ∗ + , = J = E∗ Ẽ2 Ẽ1

(66)

where ν̃1 , Ẽ1 and ν̃2 , Ẽ2 are the operators of the time-dependent Poisson’s ratios and Young’s moduli for the viscoelastic beam (target) and viscoelastic sphere (impactor), respectively. In order to deduce the integro-differential equations for the values y(t) and P(t), it is needed to prescribe the form of the operators Ẽ1 and Ẽ2 and to find the viscoelastic operators describing the time-dependent Poisson’s ratios. The operator of rigidity k̃ according to equations (65) and (66) takes the form √R 4 . k̃ = −1 2 3 (1 − ν̃1 )Ẽ1 + (1 − ν̃22 )Ẽ2−1

(67)

̃ first it is necessary to calculate the operators Ẽ (1 − ν̃2 )−1 . To find the operator k, i i Assuming the bulk modulus as a constant value, i. e., neglecting the bulk relaxation as compared with the shear relaxation, Ei∞ Ẽi = 1 − 2ν̃i 1 − 2νi∞

(i = 1, 2),

(68)

where νi∞ are the non-relaxed Poisson’s ratios, and considering (58) from equation (68) we find 1 γ ν̃i = νi∞ + (1 − 2νi∞ )νiε ∋∗γ (τiε ), 2

(69)

where νiε =

Ei∞ − Ei0 . Ei∞

Now it is needed to decode the operator (1 − ν̃i2 )Ẽi−1 . It could be done utilizing the algebra of dimensionless Rabotnov operators, the details of which can be found in [49, 52]. As a result, the rigidity operator k̃ is constructed, which allows us to obtain the equation governing the contact force P(t), i. e., 2/3

3κ ( ∞) 4√R

4

t

t − t󸀠 )P(t 󸀠 )dt 󸀠 ] [P(t) + ∑ ni ∫ ∋γ (− τi i=1 0

2/3

Fractional calculus in structural mechanics | 179 t

t

0

0

∞ 1 nπ = V0 t − ∫ P(t 󸀠 )(t − t 󸀠 )dt 󸀠 − ∑ sin( ) ∫ Gn (t − t 󸀠 )P(t 󸀠 )dt 󸀠 , m 2 n=1

(70)

as well as the equation for defining the penetration y(t), i. e., t

t󸀠

0

0

4 t 󸀠 − t 󸀠󸀠 3/2 󸀠󸀠 󸀠󸀠 4√R y(t) = − )y (t )dt ](t − t 󸀠 )dt 󸀠 ∫[y3/2 (t 󸀠 ) − ∑ mj ∫ ∋γ (− 3mκ∞ t j j=1 t

nπ 4 √R ∞ + V0 t − ∑ sin( ) ∫ Gn (t − t 󸀠 )[y3/2 (t 󸀠 ) 3κ∞ n=1 2 0

4

t󸀠

j=1

0

− ∑ mj ∫ ∋γ (−

t 󸀠 − t 󸀠󸀠 3/2 󸀠󸀠 󸀠󸀠 )y (t )dt ]dt 󸀠 , tj

(71)

where κ∞ =

2 2 1 − ν2∞ 1 − ν1∞ + . E1∞ E2∞

Since the duration of the impact interaction is rather short, for small magnitudes of t the function ∋γ (−t/τσ ), according to equation (19), could be assigned in the form ∋γ (−

t γ−1 t )≈ γ , τσ τσ Γ(γ)

(72)

whence it follows that equation (71) is reduced to t

󸀠 3/2

y(t) = V0 t − ΔV03/2 ∫[(t ) 0

t󸀠

γ−1

− δγ ∫(t 󸀠 − t 󸀠󸀠 )

3/2

(t 󸀠󸀠 )

dt 󸀠󸀠 ](t − t 󸀠 )dt 󸀠 ,

(73)

0

where δγ =

1 4 mj ∑ , Γ(γ) j=1 tjγ

Δ=

∞ 4√R nπ (1 + m ∑ An ωn cos φn sin( )). 3mκ∞ 2 n=1

Using the fugacity of the impact process, the approximate expressions for the local bearing of the target and impactor’s viscoelastic materials have been obtained for different values of viscosity, which allows us to make the following conclusion: the increase in viscosity corresponding to the variation of the fractional parameter γ from 0 to 1 results in the increase of the contact duration, as well as in the increase in the contact force maximum. This approach was used also for solving various problems on impact varying the features of the impactor (rigid, elastic, or viscoelastic) and the viscoelastic target,

180 | Y. Rossikhin and M. Shitikova namely, a Timoshenko beam with and without extension of its middle surface [56], a Kirchhoff–Love plate [61], an Uflyand–Mindlin plate considering the extension of its middle surface [62], and the low-velocity impact response of a pre-stressed isotropic Uflyand–Mindlin plate [63]. In doing so, the behavior of the Timoshenko beam and the Uflyand–Mindlin plate was analyzed in terms of one-term ray expansions in the framework of the wave theory of impact [42].

5 Non-linear dynamics problems with fractional damping In 1998, Rossikhin and Shitikova [38] focused on the application of fractional calculus in non-linear vibrations of suspension bridges under the conditions of different internal resonances, pioneering the utilization of the approximate formula Dγ eiωt ≈ (iω)γ eiωt ,

(74)

which differs from the exact formula by one term, i. e., γ iωt

D e

γ iωt

= (iω) e

sin πγ uγ −ut + e du, ∫ π u + iω ∞

(75)

0

where ω is the frequency, t is the time, i = √−1, γ (0 < γ ≤ 1) is the fractional parameter, t

1 x(s)ds d D x(t) = , ∫ Γ(1 − γ) dt (t − s)γ γ

(76)

0

Γ(1 − γ) is the gamma-function, and x(t) is an arbitrary function. The approximate formula (74), together with the non-traditional definition of the fractional derivative (see equation (5.82) in Samko et al. [67] referring to Krasnosel’skii et al. [25] and Iosida [22]) Dγ = (

γ

d ) , dt

(77)

which makes a good combination with the method of multiple time scales, allowed the authors of [38] to completely solve the stated problem. The same authors [48] succeeded in 2012 in showing that if one uses the method of multiple time scales for the analysis of non-linear vibration of mechanical systems and restricts oneself to zero- and first-order approximations, then the utilization of the approximate formula is quite reasonable, since the second term of the exact formula (75) does not enter into this approximation. This result is rather important, and

Fractional calculus in structural mechanics | 181

it cannot be ignored. Some further details will be provided considering free damped vibrations of the Duffing-like oscillator with quadratic and cubic non-linearities and damping defined by a fractional derivative ̈ + β( mx(t)

γ

d ) x(t) + k0 x(t) + k1 x(t)2 + k2 x(t)3 = 0, dt

(78)

where x, β, k0 , k1 , and k2 are, respectively, the oscillator’s displacement, the damping coefficient, the linear stiffness, and small parameters of non-linear stiffness. Dividing equation (78) by the mass m and introducing dimensionless values t ̃ = tΩ0 ,

x̃ =

x , x0

ω̃ 20 =

ω20 Ω20

(79)

yields ẍ̃ + ε2 μ(

γ

d ) x̃ + ω̃ 20 x̃ + εK̃ 1 x̃ 2 + ε2 K̃ 2 x̃ 3 = 0, dt ̃

(80)

where ε2 μ =

β γ−2 Ω , m 0

k εK̃ 1 = 1 x0 Ω−2 0 , m

k ε2 K̃ 2 = 2 x02 Ω−2 0 , m

ε is a small parameter which is of the same order of magnitude as the amplitudes, μ is a finite value, ω20 = k0 /m, Ω20 = g/l0 , x0 = mg/k0 = g/ω20 , g is the acceleration of gravity, and l0 is the spring length in the undeformed state. For ease of presentation, tildes over dimensionless values will be omitted henceforth. Equation (80) is subject to the following initial conditions: x(0) = X0 ,

̇ x(0) = V0 ,

(81)

where X0 and V0 are constant values.

5.1 Method of solution An approximate solution of equation (80) for small amplitudes weakly varying with time can be represented by an expansion in terms of different time scales in the following form: x(t, ε) = x0 (T0 , T1 , . . .) + εx1 (T0 , T1 , . . .) + ε2 x2 (T0 , T1 , . . .) + ⋅ ⋅ ⋅ .

(82)

Here, Tn = εn t (n = 0, 1, 2, . . .) are new independent variables, among them: T0 = t is a fast scale, characterizing motions with the natural frequency ω0 , and T1 = εt, T2 = ε2 t, . . . Tn are slow scales, characterizing the modulations of the amplitude and phase.

182 | Y. Rossikhin and M. Shitikova To utilize the standard procedure of the method of multiple time scales, we consider the notion that the first and second time derivatives are defined, respectively, as follows: d = D0 + εD1 + ε2 D2 + ⋅ ⋅ ⋅ , dt

d2 = D20 + 2εD0 D1 + ε2 (D21 + 2D0 D2 ) + ⋅ ⋅ ⋅ . dt 2

(83)

The fractional derivative is determined [38] as γ

(

d γ γ γ−1 ) = (D0 + εD1 + ε2 D2 + ⋅ ⋅ ⋅) = DRL0+ + εγDRL0+ D1 dt 1 γ−2 γ−1 + ε2 γ[(γ − 1)DRL0+ D21 + 2DRL0+ D2 ] + ⋅ ⋅ ⋅ , 2 γ

γ−1

(84)

γ−2

where Dn = 𝜕/𝜕Tn , and DRL0+ , DRL0+ , DRL0+ , . . . are the Riemann–Liouville fractional derivatives in time t. Substituting (82)–(84) in equation (80), equating the coefficients at equal powers of ε, and limiting ourselves by terms of ε3 , we are led to the following set of recurrent equations: D20 x0 + ω20 x0 = 0,

(85)

D20 x1 + ω20 x1 = −2D0 D1 x0 − K1 x02 ,

(86)

γ

D20 x2 + ω20 x2 = −2D0 D1 x1 − (D21 + 2D0 D2 )x0 − μDRL0+ x0 − 2K1 x0 x1 − K2 x03 ,

(87)

D20 x3 + ω20 x3 = −2D0 D1 x2 − (D21 + 2D0 D2 )x1 − 2(D0 D3 + D1 D2 )x0 γ

γ−1

− μDRL0+ x1 − μγDRL0+ D1 x0 − K1 (x12 + 2x0 x2 ) − 3K2 x02 x1 .

(88)

The general solution of equation (85) has the form ̄ 1 , T2 , T3 )e−iω0 T0 , x0 = A(T1 , T2 , T3 )eiω0 T0 + A(T

(89)

where A and Ā are yet unknown complex conjugate functions. Substituting (89) in the right-hand side of equation (86), we obtain ̄ −iω0 T0 − K1 (A2 e2iω0 T0 + 2AĀ + Ā 2 e−2iω0 T0 ). (90) D20 x1 + ω20 x1 = −2iω0 D1 Aeiω0 T0 + 2iω0 D1 Ae The functions exp(±iω0 T0 ) entering into the right-hand side of equation (90) produce secular terms, and therefore, the coefficients standing at these functions must vanish, i. e., D1 A = 0,

(91)

whence it follows that A(T1 , T2 , T3 ) = A(T2 , T3 ) is T1 -independent. Considering (91), equation (90) takes the form D20 x1 + ω20 x1 = −K1 (A2 e2iω0 T0 + 2AĀ + Ā 2 e−2iω0 T0 ).

(92)

Fractional calculus in structural mechanics | 183

Then the general solution of equation (92) has the form ̄ 1 , T2 , T3 )e−iω0 T0 + K1 ( 1 A2 e2iω0 T0 −2AA+ ̄ 1 Ā 2 e−2iω0 T0 ), (93) x1 = B(T1 , T2 , T3 )eiω0 T0 + B(T 2 3 ω0 3 where B and B̄ are yet unknown complex conjugate functions. Substituting now (89) and (93) in the right-hand side of equation (87), we are led to the following equation for determining x2 : ̄ D20 x2 + ω20 x2 = −b2 A3 exp(3iω0 T0 ) − 2K1 [A exp(2iω0 T0 ) + A]B

− [2iω0 D1 B + 2iω0 D2 A + μ(iω0 )γ A + b1 A2 A]̄ exp(iω0 T0 ) sin πγ uγ e−uT0 du − μA + cc, ∫ π u + iω0 ∞

(94)

0

where cc is the complex conjugate part to the preceding terms and b1 = 3K2 −

10K12 , 3ω20

b2 = K2 +

2K12 . 3ω20

From equation (94) it is evident that its last term does not generate secular terms and thus does not affect the solution constructed thereafter within the limits of the zero- and first-order approximations. Eliminating secular terms in (94), we obtain the solvability conditions D1 B(T1 , T2 , T3 ) = 0,

2iω0 D2 A + μ(iω0 )γ A + b1 A2 Ā = 0.

(95) (96)

From equation (95) it follows that B is T1 -independent. Let us multiply equation (96) by Ā and write its complex conjugate equation. Then we first add both obtained equations to each other, and then we subtract one from another considering that πγ πγ iγ = cos + i sin . 2 2 As a result we find ̄ 2 A − AD2 A)̄ + μωγ−1 AĀ cos πγ + b1 A2 Ā 2 = 0, i(AD 0 2 ω0 πγ ̄ 2 A + AD2 Ā + μωγ−1 AĀ sin = 0. AD 0 2

(97) (98)

Representing the function A(T2 , T3 ) in the polar form 1 A = a(T2 , T3 ) exp[iφ(T2 , T3 )], 2

(99)

184 | Y. Rossikhin and M. Shitikova from equations (97) and (98) we obtain b 1 φ̇ − δ − 1 a2 = 0, 2 8ω0

(100)

(a2 ) + sa2 = 0,

(101)

.

where an overdot denotes the T2 -derivative and γ−1

δ = μω0 cos

πγ , 2

γ−1

s = μω0 sin

πγ . 2

Integrating (100) and (101) yields 1

a = a0 (T3 )e− 2 sT2 ,

b1 2 1 a (1 − e−sT2 ) + φ0 (T3 ), φ = δT2 + 2 8ω0 s 0

(102) (103)

where a0 and φ0 are the initial magnitudes of a and φ, respectively. Thus, the zero-order approximation of the solution for non-linear free damped vibrations takes the form 1

x0 = a cos(ω0 t + φ) = a0 e− 2 αt cos[ωt + φ0 + b3 a20 (1 − e−αt )],

(104)

where α = ε2 s,

1 ω = ω0 + ε2 δ, 2

b3 =

b1 . 8ω0 s

Considering (95) and (96) and representing the function A(T2 , T3 ) and its complex ̄ 2 , T3 ) as conjugate function A(T A = c + id,

Ā = c − id,

(105)

the solution of equation (94) takes the form x2 = C(T1 , T2 , T3 )eiω0 T0 +

2K ̄ b2 3 3iω0 T0 2K1 + cc A e ABe2iω0 T0 − 21 AB + 8ω20 3ω20 ω0 T0󸀠

T0

󸀠󸀠 󸀠 sin πγ iω0 T0 − 2μ e ∫ e−2iω0 T0 dT0󸀠 ∫ f (T0󸀠󸀠 , T2 , T3 )eiω0 T0 dT0󸀠󸀠 , π

(106)

0

0

where C and C are unknown complex conjugate functions, and ∞ γ

f (T0 , T2 , T3 ) = ∫ 0

u (cu + dω0 ) −uT0 e du. u2 + ω20

(107)

Fractional calculus in structural mechanics | 185

Substituting (89), (93), and (106) in the right-hand side of equation (88), we are led to the equation for determining x3 . Eliminating the terms which produce secular terms in it, we obtain the solvability conditions D1 C(T1 , T2 , T3 ) = 0,

(108)

̄ 2 = 0. 2iω0 D2 B + μ(iω0 )γ B + 2b1 BAĀ + b1 BA

(110)

D3 A(T2 , T3 ) = 0,

(109)

From equations (108) and (109) it follows that C and A are, respectively, T1 - and T3 -independent, i. e., a0 and φ0 are constants. Let us multiply equation (110) by B̄ and write its complex conjugate equation. Then we first add both obtained equations to each other, and then we subtract one from another. As a result we find ̄ 2 B − BD2 B)̄ + μωγ−1 BB̄ cos πγ + b1 (A2 B̄ 2 + Ā 2 B2 ) + 2b1 AAB ̄ B̄ = 0, i(BD 0 2 2ω0 ω0 ̄ 2 B + BD2 B)̄ + iμωγ−1 BB̄ sin πγ + b1 (A2 B̄ 2 − Ā 2 B2 ) = 0. i(BD 0 2 2ω0

(111) (112)

Representing the function A(T2 ) and B(T2 , T3 ) in the polar form 1 A = a(T2 ) exp[iφ(T2 )], 2

1 B = b(T2 , T3 ) exp[iψ(T2 , T3 )], 2

from equations (111) and (112) we obtain b 1 1 ψ̇ − δ − 1 a2 [1 + cos 2(φ − ψ)] = 0, 2 4ω0 2 (b2 ) + sb2 + .

a2 b2 sin 2(φ − ψ) = 0, 4ω0

(113) (114)

where an overdot denotes the T2 -derivative. Considering (100), equation (113) could be rewritten as b ψ̇ − φ̇ − 1 a2 cos2 (ψ − φ) = 0, 8ω0

(115)

the solution of which due to (102) has the form tan(ψ − φ) =

b1 2 a (1 − e−sT2 ) + tan(ψ0 − φ0 ). 8ω0 s 0

(116)

Equation (114) could be rewritten as tan(ψ − φ) a2 (b2 ). = −s + , 2 2ω0 1 + tan2 (ψ − φ) b

(117)

186 | Y. Rossikhin and M. Shitikova the solution of which with due account for (102) and (116) has the form b = b0 e

− 21 sT2

b−1 1

1 + tan2 (ψ − φ) ( ) 1 + tan2 (ψ0 − φ0 )

(118)

,

where b0 and ψ0 are T3 -dependent functions as yet. However, if we carry out the expansion up to the fourth power of ε, then it can be shown that the solvability condition D3 B = 0 yields the T3 -independence of the function B, i. e., b0 and ψ0 are constants. Thus, the first-order approximation of the solution takes the form x1 = b cos(ω0 t + ψ) +

K K1 2 a cos 2(ω0 t + φ) − 12 a2 , 6ω20 2ω0

(119)

where the functions a = a(T2 ), φ = φ(T2 ), b = b(T2 ), and ψ = ψ(T2 ) are defined, respectively, by equations (102), (103), (118), and (116). The found zero-order (104) and first-order (119) approximations allow us to construct the following solution within the accuracy of ε2 : x(t, ε) = x0 (T0 , T2 ) + εx1 (T0 , T2 ) + O(ε2 ).

(120)

In order to satisfy the initial conditions (81), it is necessary that the following relationships hold: x0 (0) = X0 ,

ẋ0 (0) = V0 ,

x1 (0) = 0,

ẋ1 (0) = 0.

(121)

Substituting (104) and (119) and their derivatives with respect to time t in the initial conditions (121), we are led to the following set of equations for defining four constants a0 , φ0 , b0 , and ψ0 : X0 = a0 cos φ0 ,

(122)

1 V0 = − αX0 − a0 (ω + b3 αa20 ) sin φ0 , 2 K K b0 cos ψ0 + 12 a20 cos 2φ0 − 12 a20 = 0, 6ω0 2ω0

(123) (124)

2 ε2 a20 tan(ψ0 − φ0 ) 1 2 2 + tan (ψ0 − φ0 ) )(ω + αb a ) . tan ψ0 = (α + 3 0 2 2ω20 1 + tan2 (ψ0 − φ0 ) 1 + tan2 (ψ0 − φ0 ) −1

(125)

It could be shown [48] that if we use the Caputo fractional derivative instead of the Riemann–Liouville fractional derivative to describe the damping features, then we will obtain a solution completely equivalent to (120) within the limits of the zeroand first-order approximations. The approach described in this section, which is based on the fractional derivative definition (77) and its expansion in terms of multiple time scales (84) introduced by Rossikhin and Shitikova [38], has further been applied for solving different non-linear problems with fractional damping, including non-linear vibrations of beams, plates, shells, pipes, and suspension bridges [38, 45, 53, 59].

Fractional calculus in structural mechanics | 187

6 Conclusion In the present chapter, the viscoelastic fractional-order operators suitable for solving dynamics problems of mechanics of solids and structural mechanics have been analyzed using the examples of linear and non-linear fractional oscillators, beams, and plates, as well as impact problems, which are frequently used in engineering applications. It is shown that nowadays fractional calculus has entered the mainstream engineering analysis and is widely applied to structural dynamics problems. The place and role of the fractional calculus in engineering problems and practice have been revealed.

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HongGuang Sun, Yong Zhang, and Wen Chen

Anomalous solute transport in complex media Abstract: This chapter reviews fractional derivative models used to quantify anomalous transport of various geological materials (such as water, solutes, and sediment) in complex natural media, presents recent progress of fractional derivative diffusion equations in several application fields, and finally identifies potential challenges for fractional engines for future studies. Keywords: Anomalous transport, fractional Fick’s law, heterogeneity, fractional derivative MSC 2010: 35R11, 26A33

1 Introduction Anomalous or non-Fickian diffusion has been well observed in physical, environmental and hydrologic processes, such as water movement and contaminant transport through heterogeneous hillslopes and porous/fractured aquifers, saltwater or seawater intrusion in deep internal continents or coastal plains, and suspended load transport in rivers and streams, among many others [14]. Typical features of anomalous transport include the heavy-tailed early-time arrivals and/or late-time decline of pollutant breakthrough curves and non-Gaussian distribution of material mass density in space [30]. The study of anomalous transport is of great significance in both theoretical research (e. g., theoretical physics and statistical mechanics) and practical applications (e. g., water resource management, pollutant remediation, environment restoration, and exposure risk analysis). The study of anomalous diffusion is also of great importance in many other art and science disciplines (including chemistry, biology, and geography), engineering, and finance. In traditional approaches, Fick’s law and Darcy’s law are the basic tools to describe solute diffusive flux and water drift velocity through relatively homogeneous or macroscopically homogeneous media at almost all relevant scales. However, numerAcknowledgement: The work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2015B03814) and the National Science Funds of China (Grant Nos. 11572112, 41628202, 11528205). HongGuang Sun, Wen Chen, Institute of Soft Matter Mechanics, State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China, e-mails: [email protected], [email protected] Yong Zhang, Department of Geological Sciences, University of Alabama, Tuscaloosa AL 35487, United States, e-mail: [email protected] https://doi.org/10.1515/9783110571905-010

194 | H. G. Sun et al. ous experimental and field measurements have shown strong deviations of dynamics from those described by Fick’s law or Darcy’s law, due to the (typically multi-scale) heterogeneous nature of real-world media. For example, in natural soil or aquifers, on the one hand, inter-connected high-permeability materials can form preferential flow paths (at the scale of decimeters to a few hundred meters) and then significantly accelerate water movement and solute transport, leading to anomalous superdiffusion. On the other hand, organized, aggregated, or lumped regions with relatively low permeability and various sizes (varying from centimeters to meters) can retard flow and transport, causing anomalous subdiffusion. Anomalous transport can be explained by non-locality with spatial and/or historical dependency, which leads to the non-Gaussian distribution of chemical plumes in space and the non-Markovian evolution in time under the point source initial condition. These problems motivated the development of new theoretical models for anomalous transport in complex media, especially groundwater flow and transport, which is one of the major topics of hydrogeology [19]. Previous studies have indicated that anomalous diffusion is the typical dynamics of complex solute transport processes. Anomalous diffusion is usually defined by its mean squared displacement (MSD) using the statistical physics approach ⟨r 2 (t)⟩ ∝ t 2H ,

(1)

in which r is the radius (or distance) of diffusion, t is the elapsed time of diffusion, and H represents the Hurst parameter. It represents normal diffusion for H = 0.5, subdiffusion for H < 0.5, superdiffusion for H > 0.5, and Ballistic diffusion for H = 1.0. Three mechanisms may trigger anomalous diffusion or transport [30, 3]. First, the inherent and multi-scale heterogeneity of natural media can cause mass exchange between relatively mobile and immobile zones, retarding solute movement and resulting in a heavy-tailed concentration profile, which is typically referred to as subdiffusion. Second, the complex flow velocity field (e. g., turbulence) may yield anomalous spreading of a conservative tracer whose variance can significantly deviate from that for normal diffusion. For example, turbulence burst can be the main mechanism of suspended sediment’s superdiffusion in natural rivers. Third, chemical reactions (e. g., sorption–desorption and dissolution–precipitation) and biological activity can also cause anomalous kinetics of solutes. Anomalous diffusion can be well quantified by fractional derivative models, which are the scaling limits of continuous-time random walk (CTRW) with infinite moments. It is noteworthy that a hydrologic version of CTRW was also proposed by hydrologists [3], which is usually called the “CTRW framework”, to model anomalous transport, where various empirical memory functions are used in the general master equation and which contain the standard spatiotemporal fractional derivative models as special cases. In this CTRW framework, the particle’s trajectory is determined by

Anomalous solute transport in complex media | 195

the assumed jump length probability density function (PDF), which is typically an exponential function, and the transition time PDF (between two subsequent jumps), which can exhibit a heavy-tailed distribution. A detailed comparison between the CTRW framework and the fractional derivative model can be found in the literature [12]. As a novel modeling tool in mathematics and physics, the fractional-order derivative diffusion equation models can characterize anomalous diffusion driven by history dependence and spatial non-locality, accurately describing the tailing in breakthrough curves of solute transport. The development of fractional derivative models, as discussed below, is beyond the CTRW framework. This chapter summarizes recent progresses and discusses key challenges of fractional derivative diffusion equation models, including the research history and current development, fractional derivative models, and related applications in the field of environmental fluid mechanics.

2 Fractional derivative models for anomalous transport 2.1 Time-fractional derivative model The following time-fractional Feller diffusion equation can model subdiffusion without drift: 𝜕α c(x, t) 𝜕2 c(x, t) , = K 𝜕t α 𝜕x 2

(2)

in which K is the effective diffusion coefficient, dα /dt α is the Caputo fractional derivative, α ∈ (0, 1] is the order of the fractional derivative, and c(x, t) denotes the transition probability (which represents the concentration of solutes in aquifers, rivers, or soil). When c(x, 0) = δ(x), the MSD described by model (2) in an unbounded domain can be written as ⟨x2 (t)⟩ =

2Kt α , Γ(1 + α)

(3)

which describes subdiffusion for α ∈ (0, 1).

2.2 Space-fractional derivative model Spatial non-local transport of solutes can be caused by preferential flow paths in heterogeneous aquifers and natural soil (or fractal systems), or the velocity field mixed

196 | H. G. Sun et al. with turbulence [30]. In this case the solute concentration distribution in space is often found to be non-Gaussian and may follow Lévy distribution. Superdiffusion can be characterized by the following space-fractional derivative model: 𝜕c(x, t) 𝜕β c(x, t) =K , 𝜕t 𝜕|x|β

(4)

where 1 < β ≤ 2 is the order of the space-fractional derivative.

2.3 Time-space fractional derivative model The time-space fractional derivative model is designed to characterize the solute transport process with mixed sub- and superdiffusion due to the competition between the time memory and space non-locality, i. e., 𝜕β c(x, t) 𝜕α c(x, t) = K . 𝜕t α 𝜕|x|β

(5)

When c(x, 0) = δ(x), the MSD described by model (5) in an unbounded domain can be written as ⟨x2 (t)⟩ ∝ t 2α/β .

(6)

When 0 < 2α/β < 1, it mainly exhibits subdiffusive behavior, while superdiffusion dominates solute transport when 2α/β > 1. Especially, when α = 1 and β = 2, it reduces to normal diffusion.

2.4 Distributed-order model for multi-scale diffusion Diffusive dynamics in some porous media may evolve when crossing several spatial scales varying from micro-scale to meso-scale and macro-scale. The complex nature, and sometimes non-stationary deposits (due to the change of sediment supply) of the multi-scale heterogeneous medium, can make the single scale description unfeasible. It is necessary to investigate how and to what extent the micro-scale behavior affects the anomalous phenomena observed or measured at macro-scale. Meanwhile, in the time domain, the system may exhibit different behaviors from the short to long time ranges. Hence, the modeling approach for multi-scale diffusion and the crossscale conveying effect of solute transport have attracted wide attention in the study of anomalous diffusion. The standard fractional derivative models mentioned above fail to depict the diffusion processes involving a wide range of time spectra and/or spatial scales. Inspired by the relatively new concept of distributed-order fractional differential modeling, the

Anomalous solute transport in complex media | 197

multiple fractional derivative terms can be applied to describe multi-scale diffusion, which leads to the following governing diffusion equation: N

∑ p(α(n), t)

n=1

𝜕α(n) c(x, t) 𝜕2 c(x, t) , =K α(n) 𝜕x 2 𝜕t

(7)

where p(α(n), t) is the weight parameter and α(n) ∈ (0, 1) is the order of the timefractional derivative which characterizes different degrees of subdiffusion. The multi-term operator in (7) corresponds to different diffusive features at different time ranges. For instance, the subdiffusive feature may be measured at the micro-second scale; meanwhile, the normal diffusion behavior is observed at the hour scale. Using this model, different diffusion features at different time scales can be captured by one fractional differential equation model, and the coupling effect of multiple scales can also be reflected. This type of model can also be conveniently expanded into the Fokker–Planck equation which involves the convection of contaminants and can be stated as the following form: N

∑ p(α(n), t)

n=1

𝜕α(n) c(x, t) 𝜕c(x, t) 𝜕2 c(x, t) = −A , + K 𝜕x 𝜕x 2 𝜕t α(n)

(8)

where A is the convection coefficient. In addition, when characterizing the multi-scale superdiffusion process, we can use the following multi-term spatial fractional equation as the governing equation: N 𝜕c(x, t) 𝜕2β(n) c(x, t) = ∑ p(x, t, β(n))An , 𝜕t 𝜕|x|2β(n) n=1

0 ≤ β(n) ≤ 1,

(9)

where β(n) is the order of the spatial fractional derivative, p(x, t, β(n)) denotes the weight parameter, and An represents the convection or diffusion coefficient. The right, the normal diffusion term hand side of (9) may include the convection term 𝜕c(x,t) 𝜕x 𝜕2 c(x,t) , 𝜕x2

and/or the source term related to c(x, t). The standard spatial fractional diffusion model (with a single space-fractional derivative) is designed to characterize superdiffusion, and hence model (9) may depict multi-scale diffusion which involves both superdiffusion and normal diffusion. For instance, the feature of anomalous diffusion may be measured at the molecular scale, while the normal diffusion behavior is observed at a larger scale. In other words, different anomalous dynamics dominate at different time ranges, and the competition and coupling of multiple subdiffusion and normal diffusion described by model (9) characterize the scaling behavior well known in hydrological processes. In addition, if we replace the space-fractional term 𝜕2β(n) c(x,t) in the right-hand side of (9) by the fractional Laplacian (−Δ)β(n) c(x, t), it can 𝜕|x|2β(n) depict the multi-scale and multi-dimensional diffusion.

198 | H. G. Sun et al. It should be pointed out that, in the above models, if one of the weight parameters is much larger than the others (e. g., p(x, t, α(n)) ≫ p(x, t, α(j)) with j = 1, 2, . . . , n − 1, n + 1, . . . , N), then the above models may reduce to the diffusion models at the single scale. It means that the weighted fractional derivative terms dominate the overall dynamics of multi-scale diffusion. Model (7) has the following solution in the Fourier–Laplace domain: c(k, s) =

∑Nn=1 p(α(n))sα(n)−1

Kk 2 + ∑Nn=1 p(α(n))sα(n)

.

(10)

At the early and late times, the MSD for plumes described by model (7) grows as ⟨x 2 (t)⟩ ∝ t αmax , { 2 ⟨x (t)⟩ ∝ t αmin ,

t → 0, t → ∞,

(11)

where αmax is the maximum order of the space-fractional derivative α(⋅) and αmin is the minimum of α(⋅). If N = 1 and α(1) = 1 in (7), it reduces to the MSD of normal diffusion: ⟨x2 (t)⟩ = 2Kt. Different orders of the time-fractional derivative represent different scaling laws, and hence the multi-term model (7) leads to multiple power-law phenomena, where the power-law scaling changes with time. For example, when αmax = 1, model (7) captures the transient diffusion which shifts from normal diffusion to subdiffusion with the evolution of time.

2.5 Variable-order model for time- or space-dependent diffusion With the increasing study of anomalous diffusion in various engineering and scientific areas, the fractional derivative diffusion models have been extended. Particularly, the variable-order fractional diffusion equation model was proposed to describe anomalous diffusion varying with the elapsed time t, travel distance x, or solute concentration, especially for the analysis of solute migration in porous media. In the variableorder model, the order of the fractional derivative is a function α(x, t, f ), where f denotes the other variables that affect the diffusion process [20]. A representative expression of the variable-order diffusion model can be written as 𝜕α(x,t,f ) c(x, t) 𝜕2 c(x, t) . = K 𝜕|x|2 𝜕t α(x,t,f )

(12)

Assuming that the memory rate of the system depends only on time, the MSD for plumes described by model (12) can be written as [21] ⟨x 2 (t)⟩ =

2Kt α(t) . Γ(α(t) + 1)

(13)

Anomalous solute transport in complex media |

199

Considering the potential perturbation of the order α due to the random nature of the medium’s physical/chemical properties and solute transport, α might be a random variable which can be written as α = α0 + ϵ, where ϵ is a random number [22]. In addition, in order to explore the mechanisms underlying the variable-order fractional derivative model, other dynamic systems can also be explored. Therefore, this type of variable-order models may be called the “dynamic order” in many fractional systems. Moreover, the comprehensive study of the dynamic-order system is helpful in complex system analysis and will improve our understanding of the multi-system or multi-physical interaction.

2.6 Other types of fractional derivative models for solute transport The tempered fractional diffusion equation model, which assumes that natural dynamics are bounded with an upper limit (such as the upper limit for the random waiting times for immobile solute particles due to the finite thickness of low-permeability deposits or sorption capability of minerals), captures the complex mass transfer with a finite number of exchange rates between mobile and immobile phases, where the variation of the solute concentration in the total phase (mobile plus immobile) is due to the divergence of flux in the mobile phase and mass transfer between the two phases [13]. This model uses an exponential factor to temper the unbounded, power-law density for waiting times between particle jumps, and therefore it characterizes the bounded process occurring in nature. The solute concentration in the total phase is governed by the following fractional derivative model: 𝜕c(x, t) 𝜕α 𝜕2 c(x, t) , + βe−λt α [eλt c(x, t)] = K 𝜕t 𝜕t 𝜕x 2

(14)

in which β (β > 0) denotes the capacity coefficient and λ (λ > 0) is the tempering or truncation parameter. Recently, the discrete fractional derivative operator, a powerful tool for discrete systems, has also been used to model anomalous diffusion [25]. This model is particularly important in describing anomalous dynamics in which the physical quantity cannot be described as a continuous function. The expression of the discrete-time fractional diffusion equation can be stated as [25] Δαh c(x, t) = K

𝜕2 c(x, t) , 𝜕x 2

where Δαh c(x, t) is the discrete-time fractional derivative with the order α.

(15)

200 | H. G. Sun et al.

3 Several successful application areas of fractional derivative diffusion equation models 3.1 Solute transport in aquifers Solute transport in aquifers is one of the important research topics of groundwater hydrology and environmental sciences. Due to the complex internal architecture of natural aquifers, the hydraulic permeability can be non-uniform and anisotropic, which challenges the applicability of Fick’s law, which assumes homogeneous media. Classical differential equation models based on Fick’s law cannot reliably capture real-world dynamics. Therefore, the study of anomalous or non-Fickian diffusion in aquifers with multi-scale physical and chemical heterogeneity is practically important. The fractional diffusion equation model has been applied to analyze solute transport in various aquifers since 1998 [1]. Though it is not the first non-local transport model to characterize solute transport in complex media, the last two decades have witnessed dramatic progress in the method by the fractional derivative model. Benson et al. [2] used a one-dimensional fractional-order advection-dispersion equation model to analyze the migration of bromide ions in a fractured aquifer. Cushman and Ginn [7] found that the spatial fractional diffusion equation model with constant parameters is a special case of the non-local dispersive constitutive theory. Schumer et al. [16] applied the generalized Taylor series expansion to build the one-dimensional spatial fractional convection-dispersion equation model with constant parameters. The follow-up study by Schumer et al. [17] built the fractal mobile-immobile fractional diffusion model, by replacing the single mass transfer rate by a power-law memory function in the traditional mobile-immobile zone model. Huang et al. [11] employed the space-fractional derivative, one-dimensional advection-dispersion model to describe the migration of pollutants in fractured soil. Zhang et al. [30] reviewed previous applications of space- and time-fractional diffusion models, and identified the appropriate fractional derivative models for laboratory and field experimental data for contaminant transport. Multi-dimensional fractional derivative models were also developed and applied for mixed sub- and superdiffusion with direction-dependent scaling rates along arbitrary directions in real-world porous and fractured media [28]. Time- and space-dependent and multi-scale solute transport remains the key issue in the research of subsurface solute transport. If we consider the exchange and phase transition between mobile and immobile phases in porous media, a two-scale fractional derivative model can be obtained, such as the diffusion equation model [17]. This model can be regarded as a special case of the distributed-order derivative diffusion model, and its further expansion is called the tempered fractional derivative diffusion model [13]. Reeves et al. [15] analyzed the anomalous solute diffusion behavior in discrete fracture networks by Monte Carlo simulations, and further described the resulting transport dynamics using the α-stable statistics. Meanwhile, the

Anomalous solute transport in complex media

| 201

multi-scale fractional diffusion model can be simulated by the Lagrangian solver [29]. Sun et al. [23] also analyzed the physical mechanism of the variable-order fractional derivative diffusion model, and further employed it to accurately describe the timedependent solute migration process in aquifers.

3.2 Sediment transport Sediment transport is one historical and challenging topic in hydrology. It is the core of rock cycles and one of the major mechanisms by which the Earth’s topography is built. Its investigation mainly focuses on suspended sediment transport and sedimentation. Suspended sediment moving in the vertical direction is mainly determined by the competition between the downward settling velocity (due to gravity) and the upward motion (caused by turbulent diffusion). The two factors result in the sedimentation or erosion of the river bed/bank, and also affect the migration of pollutants as well as the ecological protection of the river. The process of bedload transport can be more complicated due to the complex velocity field near the river bed and the interaction between sediment, the open channel, and the (migrating) river bed. In bedload transport, sediment particles may be buried below the active layer of the river bed, or experience an accelerating movement due to turbulence breaking the sediment cluster. The complexity of sediment dynamics comes from the complexity of the flow velocity and the interaction between the river bed and the channel flow. The widely existing multiscale mass exchange makes the classical diffusion theory unsuitable to describe the turbulent diffusion behavior of bedload particles. The fractional derivative, stochastic differential equation model has been applied to analyze sediment transport, and some valuable results have been obtained. In the analysis of sediment’s vertical distribution, the diffusion of sediment is generally treated as normal diffusion, but the existence of turbulence makes the turbulent diffusion of sediment substantially different from normal diffusion. Previous results showed that turbulence diffusion is a typical anomalous diffusion behavior, and the fractional derivative diffusion equation model can accurately describe this type of diffusion process [26]. Based on the turbulent diffusion analysis, Chen et al. [6] proposed a new vertical distribution formula for sediment concentration by establishing a fractional derivative model associated with the properties of the Mittag-Leffler function. Comparison between the experimental data and several existing models indicated that the new formula is attractive since it has fewer model parameters and leads to accurate descriptions. The parameter sensitivity analysis showed that the fractional derivative order is the key parameter of the model. The analysis result of experimental data of several groups showed that the fractional derivative increases with the particle size. Hence, anomalous diffusion occurs more with fine particles. Bedload transport is affected by many factors, including the quantity of sediment, the river bed structure, the flow velocity, and turbulence diffusion, which can cause

202 | H. G. Sun et al. strong path dependency and spatial non-locality for sediment movement. Meanwhile, the history-dependent and non-local characteristics of fractional derivatives make the fractional stochastic differential equation model an excellent tool to accurately characterize bedload dynamics. The sediment particle may be blocked or buried by river bed structures. Hence, using the random motion approach, the probability density function of the waiting time exhibits obvious power-law tailing characteristics, while the coarsening and cluster structure of the river bed force part of the sand particles to experience a dramatically fast movement, causing the non-Gaussian distribution of sediment jump lengths. Therefore, the fractional derivative is a natural candidate to quantify the random motion of bedload sediment. Recently, fractional derivative models are gradually being used to characterize bedload transport. For example, Hill et al. [10] found experimentally and theoretically that the random jump distribution function of a single particle is an exponential function, after assuming that the river bed is composed of uniform particles. However, the random jump distribution function is a power-law distribution when the river bed is composed of mixed-size sediment, and the fractional derivative is an excellent tool for this situation. Ganti et al. [9] conducted a numerical simulation research to investigate the physical model for bedload transport under different conditions. Their results showed that the random jump distribution is close to the exponential distribution under the stable flow and near-uniform sand conditions, and the traditional convection-diffusion equation can well describe the sediment dynamics. However, since the unstable flow field and mixed particles are usually observed in natural rivers, the statistical distribution of stochastic motion can be better described by the power-law distribution. Hence, the fractional derivative model might be a better model. Bradley et al. [5] applied the spatial fractional convection-diffusion model to describe the bedload transport in a natural river under instantaneous source condition. Their simulation result showed that the spatial distribution of sediment is close to the α stable distribution, and the spatial fractional model agrees well with the experimental data. Zhang et al. [32] considered the influence of the velocity field on random motion of bedload transport along sand beds, and then established a fractional advection-dispersion equation with fewer parameters to accurately describe the complex dynamics for bedload transport in real-world rivers.

3.3 Applications in other fields Hillslope subsurface storm flow and solute transport with surface run-off are also important hydrologic applications of the fractional-order diffusion models. Hillslope flow is affected by topography, soil structure, and hydraulic properties, where water flow and solute transport can exhibit anomalous diffusion characteristics. Schumer et al. [18] introduced the power-law tailing distribution of random jumps and waiting time based on the analysis of the instability of complex velocity fields in rivers or

Anomalous solute transport in complex media

| 203

slope flows, and they further derived a fractional derivative model to describe the experimental data of solute transport with surface run-off. Foufoula-Georgiou et al. [8] characterized the non-locality of solute transport in the slope flow using the spacefractional convection-diffusion equation model, and their results showed that the sediment flux may be different even if the local slope gradient is the same. Furthermore, the slope flow is influenced by the microscopic topography, the local slope angle, and the hydraulic properties of the soil (such as the hydraulic permeability), which may cause both the preferential flow and the delayed flow, a complex response of subsurface stormflow to precipitation that can be well captured by the tempered fractional derivative model [31, 27]. Moreover, solute transport in groundwater may also be accompanied by chemical reactions or biological effects. The multi-system analysis which combines anomalous diffusion, chemical reactions, and biological activity is a research frontier in the application field [4, 24].

4 Closing remarks Generally speaking, fractional derivative models have achieved great success in characterizing anomalous transport in complex media. There are however several problems which have not been well solved. First, the predictability of parameters (especially the fractional derivative order) for the fractional solute transport equation model and the mechanism analysis remain the key problems in field applications. Second, previous investigations of anomalous diffusion in different media indicated that the statistical descriptions of different diffusion processes are different, their physical mechanism is still unclear, and the link between fractional diffusion equation models and statistical descriptions still needs further investigations. Third, the introduction of a fractional derivative in the mass conservation model leads to fractional dimensions for model parameters, whose definition and field measurement remain obscure.

Bibliography [1] [2] [3] [4]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph. D. dissertation, Univ. of Nev., Reno, 1998. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403–1412. B. Berkowitz, S. Emmanuel, and H. Scher, Non-Fickian transport and multiple-rate mass transfer in porous media, Water Resour. Res., 44 (2008), W03402. D. Bolster, P. Anna, and D. A. Benson et al., Incomplete mixing and reactions with fractional dispersion, Adv. Water Resour., 37 (2012), 86–93.

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[27] Y. Zhang, B. Baeumer, L. Chen, D. M. Reeves, and H. G. Sun, A fully subordinated linear flow model for hillslope subsurface stormflow, Water Resour. Res., 53 (2017), 3491–3504. [28] Y. Zhang and D. A. Benson, Lagrangian simulation of multidimensional anomalous transport at the MADE site, Geophys. Res. Lett., 35 (2008), L07403, 10.1029/2008GL033222. [29] Y. Zhang, D. A. Benson, M. M. Meerschaert et al., Random walk approximation of fractional-order multiscaling anomalous diffusion, Phys. Rev. E, 74 (2006), 026706. [30] Y. Zhang, D. A. Benson, and D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561–581. [31] Y. Zhang, L. Chen, D. M. Reeves et al., A fractional-order tempered-stable continuity model to capture surface water runoff, J. Vib. Control, 22(8) (1993–2003), 2016. [32] Y. Zhang, R. L. Martin, D. Chen et al., A subordinated advection model for uniform bedload transport from local to regional scales, J. Geophys. Res., Earth Surf., 119 (2014), 2711–2729.

Behrouz Parsa Moghaddam, Arman Dabiri, and José António Tenreiro Machado

Application of variable-order fractional calculus in solid mechanics Abstract: This chapter discusses the adoption of fractional derivative operators in modeling viscoelastic materials and their creep behavior. The damper elements in three conventional rheological models are replaced by fractional-order dashpots that include fixed- and variable-order fractional derivative operators. It is shown that the creep response of the presented fractional rheological models can be realistic if and only if they have monotonically decreasing variable-order functions. Finally, the presented rheological models are used to model the creep behavior of an epoxy by using experimental isothermal creep tests. Keywords: Fractional calculus, variable-order fractional derivatives, rheological model, creep, viscoelastic MSC 2010: 26A33, 34A08, 76A10, 74H15

1 Introduction Fractional integral and derivative operators are extensions of the integer-order integral and derivative operators in the classic calculus so that the orders are relaxed to include real or complex numbers. It has been shown that fractional derivative operators are useful in describing dynamical processes with memory or hereditary properties such as creep or relaxation processes in viscoelastoplastic materials [11, 14, 26], bioengineering [23], impact problems [8], diffusion process models [12, 40], plasma physics [15], and control problems [9, 24, 18, 19]. Fixed-order fractional derivative operators are defined such that their orders are constant. Recently the definition of fixedorder integral and derivative operators has been extended to include functional orders. Moreover, they were initially defined by extending the definition of fixed-order fractional integral and derivative operators in the sense of Riemann–Liouville (RL) [41, 39]. There are more definitions for variable-order fractional (VOF) derivative op-

Behrouz Parsa Moghaddam, Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran, e-mail: [email protected] Arman Dabiri, School of Engineering Technology, Eastern Michigan University, Michigan, USA, e-mail: [email protected] José António Tenreiro Machado, Department of Electrical Engineering, Institute of Engineering, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal, e-mail: [email protected] https://doi.org/10.1515/9783110571905-011

208 | B. Parsa Moghaddam et al. erators (VOFDOs) such as the RL [39], Coimbra [7, 48], Grünwald [47], and Lorenzo– Hartley [22] definitions. Since the kernel of VOFDOs has variable exponents, obtaining an analytical solution for fractional differential equations including VOFDOs is usually difficult or impossible [44, 51, 3, 4, 27]. This issue is one of the main limitations of employing VOFDOs in practical applications. The most common numerical algorithms for solving VOF differential equations (VOFDEs) are based on finite-difference [49, 43, 6, 42, 54, 28, 52, 29, 30, 32, 25, 20] and spectral [53, 5, 2] methods. In [17], several numerical techniques were proposed to approximate the solution of VOFDOs. Furthermore, the presented numerical schemes were used to obtain a proper fit between the experimental data and the response of mathematical models in deformations and vibrations of viscoelastic materials. In [29], a stable three-level explicit spline finite-difference algorithm was suggested based on a linear B-spline approximation of the VOFDO. The proposed approximations were used for solving a class of VOF partial differential equations. Fixed- and variable-order fractional integral and derivative operators can be used in characterizing hereditary properties of dynamical systems with uniform and nonuniform memory, respectively. VOFDOs were used to characterize the dynamic of Van der Pol oscillators [13, 31], describe biochemical tumorous bone remodeling models [33], analyze elastoplastic indentation problems [16], model the behavior of systems with multiple fractional terms [48], formulate a model in the scope of statistical mechanics [37], and develop control laws [10, 9, 35, 34]. Furthermore, the advantages of using VOFDOs were discussed for describing the behavior of a mass-spring-damper system with a damper element composed of a non-linear viscoelastic material [7]. In [50], the pros and cons of using different types of VOFDOs were shown in a comparative analysis of anomalous relaxation processes. Further details about applications of VOFDOs can be found in [27, 29, 17, 33, 50, 38, 21, 45, 46]. This chapter reviews the application of fixed- and variable-order fractional derivative operators in the modeling of linear viscoelastic materials and compares different models with distinct types of VOFDOs. Three VOF rheological models are constructed by replacing the damper elements in regular rheological models with VOF dashpots. The effects of including VOF dashpots modeled by different VOFDs on the creep response of the VOF rheological models are numerically studied. Next, the effects of different types of VOFDOs for VOF models are discussed. Finally, the advantages of using the proposed models are shown in modeling the creep response of an epoxy by using experimental data. The chapter is organized as follows. Section 2 overviews the fundamental concepts in fractional calculus. Section 3 presents the discretization scheme of VOFDOs for the numerical analysis of VOF models. Section 4 recalls the theory of linear viscoelasticity along with three classical rheological models, namely the Kelvin–Voigt, Maxwell, and Zener models. Section 5 introduces three VOF rheological models. Section 6 analyzes fixed- and variable-order fractional creep constitutive equations for an epoxy using experimental data. Finally, Section 7 outlines the main conclusions.

Application of variable-order fractional calculus in solid mechanics | 209

2 Preliminary concepts In this chapter, the VOFDO with a strong memory proposed by Lorenzo and Hartley is used [22]. Definition 1. Let Re(α(⋅, ⋅)) ∈ ℝ+ . The VOF integral for y(t) ∈ ℝ is defined by ν(⋅) α(⋅,⋅) y(t) 0 Jt

t

=∫ 0

1 (t − ζ )α(t,ζ )−1 y(ζ )dζ , Γ(α(t, ζ ))

(1)

where ζ is an auxiliary variable that belongs to the interval [0, t] and Γ(⋅) is the Gamma function. Definition 2. Let α(⋅, ⋅) ∈ ℝ+ , m − 1 < α(⋅, ⋅) < m, m ∈ ℕ, y(t) be (m − 1) times continuously differentiable, and let y(m) (t) be once integrable. The VOF derivative of y(t) is defined by ν(⋅) α(⋅,⋅) y(t) 0 Dt

t

=∫ 0

y(m) (ζ )dζ 1 . Γ(m − α(t, ζ )) (t − ζ )α(t,ζ )+1−m

(2)

Definition 3. Consider the following forms for α(t, ζ ) in Definition 2: α(t, ζ ) := α(t), α(t, ζ ) := α(ζ ),

m − 1 < α(t) ≤ m, m − 1 < α(ζ ) ≤ m,

(3)

and in the follow-up we shall call them “variable-order function” or briefly “VOfunction”. Replacing α(t, ζ ) in equation (2) with the forms in equation (3) results in obtaining the types 1 and 2 of VOFDOs denoted by ν1-type and ν2-type, respectively. Consequently, their definitions are t

ν1 α(t) 0 Dt y(t)

ν2 α(t) 0 Dt y(t)

=∫ 0

t

=∫ 0

y(m) (ζ )dζ 1 , Γ(m − α(t)) (t − ζ )α(t)+1−m

(4a)

y(m) (ζ )dζ 1 . Γ(m − α(ζ )) (t − ζ )α(ζ )+1−m

(4b)

Lemma 1 ([1]). Let α, β > 0, λ ∈ ℝ, and pα > |λ|. Then we have −1

ℒ {

pα−β } = z β−1 Eα,β (−λz α ), pα + λ

(5)

where ℒ−1 {⋅} denotes the inverse Laplace transform operator and Eα,β (z) is the generalized Mittag-Leffler function defined by zk , Γ(β + αk) k=0 ∞

Eα,β (z) = ∑

Re{α} > 0, z ∈ ℂ.

(6)

210 | B. Parsa Moghaddam et al.

3 Linear B-spline approximation schemes In this section, the linear B-spline approximation (SM-algorithm) proposed in [32, 25] is used to approximate the VOFDOs in equation (4). Let the interval Ω = [0, T] be divided into {t0 , t1 , . . . , tN }, t0 = 0 < t1 < ⋅ ⋅ ⋅ < tN = T, with a uniform step size h, h ∈ ℝ+ , such that tn = nh, n = 0, 1, . . . , N. Proposition 1 ([32]). Let y(t) ∈ C m+1 (Ω) and m − 1 < α(t) ≤ m, m ∈ ℕ. A linear B-spline approximation of the type 1 VOFDO (4a) is ν1 α(⋅) 0 Dtn y(t)

n

hm−α(tn ) y(m) (tk )an,k , Γ(m + 2 − α(t )) n k=0

(7)

≈ ∑

where an,k

(n − 1)m+1−α(tn ) − nm−α(tn ) × (n − (m + 1 − α(tn ))), { { = {(n − k + 1)m+1−α(tn ) − 2(n − k)m+1−α(tn ) + (n − k − 1)m+1−α(tn ) , { {1,

k = 0, 1 ≤ k ≤ n − 1, k = n.

Proposition 2 ([25]). Let y(t) ∈ C m+1 (Ω) and m − 1 < α(t) ≤ m. Then, a linear B-spline approximation of its type 2 VOFDO is ν2 α(⋅) 0 Dtn y(t)

n

hm−α(tk ) y(m) (tk )bn,k , Γ(m + 2 − α(t )) k k=0

≈ ∑

(8)

where sup{α(t)|m − 1 < α(t) ≤ m, t ∈ Ω}, α∗ = { inf{α(t)|m − 1 < α(t) ≤ m, t ∈ Ω},

if tn ≤ 1, if tn > 1.

Proposition 3 ([32]). Let m − 1 < α(t) ≤ m and y(t) ∈ C m+1 (Ω). Then, there exists an α∗ -dependent constant, i. e., Cα∗ > 0, such that the truncated error of the VOFDO obtained by the SM-algorithm satisfies ∗ 󵄨 󵄨󵄨ν(⋅) α(t) (m+2) ν(⋅) α(t) (η)‖∞ h2+m−α , 󵄨󵄨0 Dtn y(t) − (0 Dtn y(t))SM 󵄨󵄨󵄨 ≤ Cα∗ ‖y

(9)

where η is an arbitrary value that belongs to Ω and sup{α(t)|m − 1 < α(t) ≤ m, t ∈ Ω}, α∗ = { inf{α(t)|m − 1 < α(t) ≤ m, t ∈ Ω},

if tn ≤ 1, if tn > 1.

Since y(m) (tk ), k = 0, . . . , n, are not usually available for VOFDEs with boundary conditions, a central differential operator can be used to solve them. The central difference approximation of the mth-order derivative of y(t) is given by y(m) (tk ) =

1 δ (t ) + 𝒪(h2 ), hm m k

(10)

Application of variable-order fractional calculus in solid mechanics | 211

where m m m δm (tk ) = ∑ (−1)i ( ) y(tk + ( − i)h) i 2 i=0

(11)

and 𝒪(⋅) denotes the convergence order of approximation. The following illustrative example shows the effects of employing the two VOFDOs in equation (4) on calculating VOF derivatives of the continuous function y(t) = t cos(2πt) for 0 < α(t) ≤ 1. The analytical solution of the ν1-type VOF derivative of the order α(t) of y(t) is ν1 α(⋅) 0 Dt y(t)

=

1 ((2π)α(t)+0.5 t √tα(t)s0.5−α(t),1.5 (2πt) Γ(2 − α(t)) − (2π)α(t)−0.5 √ts1.5−α(t),0.5 (2πt)(α(t) + 1) + t 1−α(t) ),

(12)

where sμ,ν (t) denotes the Lommel function defined as sμ,ν (t) = t

1 μ+1 1 F2 ([1]; [ 2 (μ

2

− ν + 3), 21 (μ + ν + 3)]; −t4 ) (μ + 1)2 − ν2

,

(13)

in which p Fq (a1 , . . . , ap ; b1 , . . . , bq ; t) denotes the generalized hypergeometric function. Its ν2-type VOF derivative of order α(t) is obtained by means of the SM-algorithm. Without loss of generality let α(t) = t. Figure 1 shows the ν1-type and ν2-type VOF derivatives of y(t) using the SM-algorithm with h = 0.005, α(t) = t, and t ∈ [0, 1]. It is shown that the ν1-type VOF derivative curve starts from 0 and intersects the fixedorder fractional derivative curve of the order αk at the time t = αk , k = 0, 1, . . . , 10. In other words, the ν1-type VOF derivative response of a function at the time t = T, 0 ≤ T ≤ 1, is the response of the fixed-order fractional derivative of order α(T) at that time. However, this is not true for the ν2-type VOF derivative solution.

Figure 1: The fixed-order fractional derivatives of y(t) = t cos(2πt) with α = {0.0, 0.1, 0.2, . . . , 0.9, 1} and ν1-type (black solid line), ν2-type (black dashed line) VOF derivatives of y(t) = t cos(2πt) when α(t) = t in the interval [0, 1].

212 | B. Parsa Moghaddam et al.

4 Linear viscoelasticity In viscoelastic materials, the relationship between stress and strain is only dependent on time. Three common experiments that are used to obtain the relation between stress and strain are ramp, creep, and relaxation tests. In a ramp test, the force is increased continuously within a certain time, and the resulting deformation is measured over time. In a creep test, the strain response to a unit step of stress is obtained.

Figure 2: A typical creep curve with the loading, primary, secondary, and tertiary stages.

In a relaxation test, the stress response to a unit step of strain is obtained. A typical creep curve has four stages: the loading, primary, secondary, and tertiary stages, as shown in Figure 2. In the loading regime, the load is ramped up to a desired stress, and then it is kept constant. This stage is followed by the primary stage, where strain hardening occurs and the strain rate decreases. Subsequently, the material goes to the secondary regime where the strain rate is nearly constant in the entire deformation. Finally, in the tertiary regime, the material begins to yield and the strain rate increases rapidly. The response of the material in a creep test with a constant stress is described by creep compliance J(t). The creep compliance is defined as the change in the strain ε(t) due to the applied constant stress σ0 , i. e., ε(t) = J(t)σ0 .

(14)

The creep compliance quantifies the behavior of the material to an applied unit stress. Due to the natural form of the independent solutions of linear ordinary differential equations in the form of the exponential function exp(−t/τ), the creep compliance can be described as n

J(t): = ∑ ji exp(−t/τi ), i=0

(15)

where τi are called relaxation times and ji ∈ ℝ+ . The unknown constants ji and the relaxation times are determined through a fitting procedure with the experimental data.

Application of variable-order fractional calculus in solid mechanics | 213

Similarly, the response of the material in a relaxation test is described by the relaxation modulus E(t). The relaxation modulus is defined as the change in the stress σ(t) due to the applied constant stress ε0 , i. e., σ(t) = E(t)ε0 .

(16)

The constitutive equation of a linear viscoelastic material can be defined by using the Boltzmann superposition principle and causal histories as t

t

ε(t) = ∫ J(t − τ)dσ(τ) = σ(0 )J(t) + ∫ J(t − τ)σ 󸀠 (τ), +

0 t

0− t

(17)

σ(t) = ∫ E(t − τ)dε(τ) = ε(0 )E(t) + ∫ E(t − τ)ε (τ). +

󸀠

0

0−

In addition, the instantaneous (t → 0 ) and long-term (t → +∞) behavior of a material can be defined by its creep compliance and relation modulus defined by the glass compliance Jg : = limt→0+ J(t), the glass modulus Eg : = limt→0+ E(t), the equilibrium compliance Je : = limt→+∞ J(t), and the equilibrium modulus Ee : = limt→+∞ E(t). According to the definition of the creep compliance and the relation modulus and using the constitutive equation (17), one can show the reciprocity relation ℒ{J(t)}ℒ{E(t)} = 1/p2 , where ℒ{⋅} and p denote the Laplace transform and variable, respectively. The constitutive equation (18) can be solved for strain (or stress) by using the known stress (or strain) over time. Furthermore, the behavior of linear viscoelastic materials can be described by linear differential equations. In general, a constitutive equation for a linear viscoelastic material is given by +

𝒜σ = ℬε,

(18)

i where 𝒜: = ∑ni=0 ai Dti , ℬ: = ∑m i=0 bi Dt , ai , bj ∈ ℝ, i = 0, 1, . . . , n, j = 0, 1, . . . , m, and n ≥ m. Thus, models including connected common mechanical elements such as springs and dampers can be used to visualize the constitutive equation in a convenient way. These descriptions are known as rheological models constructed by combining linear springs and dampers in series and parallel. Figure 3 shows three well-known rheological models, called Maxwell, Kelvin–Voigt, and Zener models. More complex rheological models with more realistic responses can be constructed by including additional elements. A Kelvin–Voigt element is composed of a linear spring and a damper connected in parallel, and its constitutive equation is given as

ηDt ε(t) + Eε(t) = σ(t),

(19)

where E is the elasticity modulus and η is the viscosity. Under a creep test with σ(t) = σ0 and ε(0) = 0, the response is obtained as ε(t) = (1 − exp(−E/ηt))

σ0 . E

(20)

214 | B. Parsa Moghaddam et al.

Figure 3: Three well-known rheological models: (a) Kelvin–Voigt model; (b) Maxwell model; and (c) Zener model. η

The single relaxation time of the creep compliance is τ1 = E . Increasing the damping coefficient η or decreasing E causes a faster response in converging the strain to the steady value. A Maxwell element is composed of a linear spring and a damper connected in series, and its constitutive equation is given as Dt ε(t) =

Dt σ(t)

E

+

σ(t) . η

(21)

Under a creep test with σ(t) = σ0 and ε(0) = 0, the response is ε(t) = (

1 1 + t)σ0 . E η

(22)

It is noticed that the creep response of this model does not have an exponential behavior. It linearly changes over time, which is not a realistic behavior for many viscoelastic materials. A Zener element is composed of a linear spring and a Maxwell element connected in parallel, and its constitutive equation is given as Dt ε(t) +

E2 E1 E2 1 ε(t) = D σ(t) + σ(t). E1 + E2 E1 + E2 t η(E1 + E2 )

(23)

Under a creep test with σ(t) = σ0 and ε(0) = 0, the response of the model is ε(t) = (1 − exp(−

σ E1 E2 t)) 0 . E1 + E2 ηE1

(24)

The creep compliance has, again, a single relaxation time τ1 = E1 + E1 . 1 2 The creep compliance of all the rheological models represented by ordinary differential equation (18) is in the form of an exponential function (equation (15)). However, in several experiments, it has been shown that the creep compliance does not usually have exponential behavior, and its behavior is more like a power-law response [36], i. e., n

J(t): = ∑ ji (−t/τi )−iα , i=0

(25)

Application of variable-order fractional calculus in solid mechanics | 215

where α ∈ ℝ+ . Therefore, it is necessary to modify these rheological models in order to satisfy the above fact. One way to solve this problem is using VOFDOs, which is explained in the next section.

5 VOF rheological models One of the main issues for employing VOFOs in rheological models is determining a proper VO-function for VOFEOs. A VO-function for rheological models is usually chosen such that they are bounded between 0 and 1. In this section, it is shown that VOfunctions in rheological models should be monotonically decreasing. A VOF dashpot can be interpreted as a linear spring with the stiffness coefficient η, or a linear damper with the damping coefficient η, if α(t) = 0 or α(t) = 1, respectively. If 0 < α(t) < 1, then a unique behavior is obtained for VOF dashpots that cannot be determined by any combination of linear springs or damper elements. Moreover, from a physical perspective, a linear spring is a perfect energy storage mechanism, while a linear damper dissipates the energy. A VOF dashpot in a viscoelastic model adds flexibility in tuning the energy dissipation rate by adjusting its order [8]. VOF rheological models are constructed by replacing the dampers or the springdamper elements of the classical rheological models with VOF dashpots. Dashpots are fractional dampers that satisfy the force-displacement relation α(t) Fc (t) = η ν(⋅) 0 Dt x(t),

(26)

where η is the fractional damping coefficient and α(t) ∈ [0, 1]. Figure 4 shows VOF Maxwell, Kelvin–Voigt, and Zener models. The constitutive equation of the VOF Kelvin–Voigt model is obtained by replacing the damper term ηDt ε(t) in the conα(t) stitutive equation (19) with a VOF dashpot term η ν(⋅) 0 Dt ε(t), yielding α(t) Eε(t) + η ν(⋅) 0 Dt ε(t) = σ(t).

(27)

Figure 4 illustrates that a spring and a VOF dashpot are connected in the VOF Maxwell α(t) model such that σE (t) = ElE , ση (t) = η ν(⋅) 0 Dt ε(t), and σ(t) = σE (t) = ση (t). If we consider that L = lE + lη and if we apply VOF differentiation of order α(t) to both sides, then the constitutive equation of the VOF Maxwell model is obtained as ν(⋅) α(t) 0 Dt ε(t)

=

ν(⋅) α(t) 0 Dt σ(t)

E

+

σ(t) . η

(28)

It is noticed that equation (28) is similar to equation (21) whose first-order derivatives are replaced with FD. Similarly, it can be shown that the constitutive equation of the VOF Zener model is obtained similar to equation (23) such that integer-order derivaα(t) tives are replaced by ν(⋅) 0 Dt [⋅].

216 | B. Parsa Moghaddam et al.

Figure 4: Three VOF rheological models: (a) VOF Kelvin–Voigt model; (b) VOF Maxwell model; and (c) VOF Zener model.

In general, consider the following VOFDE: ν(⋅) α(t) 0 Dt ε(t)

+ c1 ε(t) = c2

ν(⋅) β(t) 0 Dt σ(t)

+ c3 σ(t),

(29)

where ci ∈ ℝ+ , i = 1, 2, 3. The constitutive equation of the proposed VOF rheological models can be obtained by adjusting the parameters ci in the VOFDE (29). For example, η (i) the constitutive equation of the VOF Kelvin–Voigt model is obtained when c1 = E , 1 c2 = 0, c3 = E , and β(t) = α(t); (ii) the constitutive equation of the VOF Maxwell model is obtained when c1 = 0, c2 = E1 , c3 = η1 , and β(t) = α(t); (iii) the constitutive equation

E2 1 2 , and , c2 = E +E , c3 = η(EE+E of the VOF Zener model is obtained when c1 = EE1+E 1 2 1 2 1 2) β(t) = α(t). All the previous settings for ci yield the classic rheological models when α(t) = β(t) = 1. Since the creep response of the VOF Zener model is similar to the one exhibited by the VOF Kelvin–Voigt model, the creep response of the VOF Kelvin–Voigt model is studied only for the different VO-functions. There is no analytical solution of the VOFDE (29) in the case of having time-dependent VO-functions. Therefore, VOFDE (29) must be numerically solved by using the SM-algorithm presented in Section 3. The following two types of VO-functions are considered for the variable orders to study the creep response of the VOF rheological models: 1. monotonically increasing VO-functions: α(t) = β(t) = c4 exp(−c5 t) and α(t) = c α(t) = 1+c4 t 2 , 0 ≤ c4 , c5 ≤ 1; 5 2. monotonically decreasing VO-functions: α(t) = β(t) = c4 (1 − exp(−c5 t)) and α(t) = α(t) = c4 (1 − 1+c1 t 2 ), 0 ≤ c4 , c5 ≤ 1. 5

In this chapter, since the creep behavior of the ν2-type rheological model is qualitatively similar to that of the ν1-type rheological model, they are skipped for parsimony. In the first case, let α(t) = α0 , 0 ≤ α0 ≤ 1. Since the order is time-invariant, this rheological model is called fixed-order fractional Kelvin–Voigt model. Using the Laplace transform and Lemma 1 yields the creep response to a constant stress σ(t) = σ0 as ε(t) = t α0 Eα0 ,α0 1 (−c1 t α0 )c3 σ0 .

(30)

Application of variable-order fractional calculus in solid mechanics | 217

Figure 5: The creep behavior of the fixed-order fractional Kelvin–Voigt model for different values of E = {0.3, 0.5, 0.8, 1} and α0 = {0.3, 0.4, . . . , 0.9, 1} when η = 0.5 and σ0 = 1.

Equation (30) shows that the power-law response for the creep compliance (equation (25)) can be partially compensated by the use of VOFOs. Moreover, Figure 5 shows the creep behavior of the fixed-order fractional Kelvin–Voigt model for different values of E = {0.3, 0.5, 0.8, 1} and α0 = {0.3, 0.4, . . . , 0.9, 1} when η = 0.5 and σ0 = 1. It is shown that decreasing E or α0 decreases the settling time. The fractional order α0 has significant effects on the primary stage of the creep curves which is due to the power term in the solution. In addition, changing η and E results in similar behavior in the creep response for both the VOF Kelvin–Voigt and the Kelvin–Voigt models. In the second case let α(t) = c4 exp(−c5 t), which is a monotonically decreasing VOfunction. Figure 6 shows the creep response of the VOF Kelvin–Voigt model for different values of c4 = {0.3, 0.4, . . . , 0.9, 1} and c5 = {0.1, 0.5, 0.7, 1} when η = 1, E = 1, and σ0 = 1. The coefficient c4 works similarly to the order of the fixed-order Kelvin– Voigt model and significantly affects the initial response of the creep curves. On the other hand, the coefficient c5 mainly affects the shape of the secondary creep stage. It is noticed that large values of c4 and c5 give unrealistic creep responses due to the

218 | B. Parsa Moghaddam et al.

Figure 6: The creep behavior of the VOF Kelvin–Voigt model with ν1-type VOFDOs for different values of c4 = {0.3, 0.4, . . . , 0.9, 1} and c5 = {0.1, 0.4, 0.7, 1} when α(t) = c4 exp(−c5 t), η = 1, E = 1, and σ0 = 1.

Figure 7: The creep behavior of the fractional Kelvin–Voigt model with ν1-type fractional operators for different values of c4 = {0.3, 0.4, . . . , 0.9, 1} and c5 = {0.4, 1} when α(t) = c4 (1 − exp(−c5 t)) and η = E = σ0 = 1.

Application of variable-order fractional calculus in solid mechanics | 219

decreasing of the strain in the secondary stage (i. e., having a negative strain rate). Therefore, the values of c4 and c5 should be chosen small such that the strain rate is c kept positive in the entire creep process. Changing the VO-function to α(t) = 1+c4 t 2 re5 sults in the same conclusion, and consequently, they are not presented here. In the third case, let α(t) = c4 (1 − exp(−c5 t)) represent a monotonically increasing function. Figure 7 shows the creep response of the VOF Kelvin–Voigt model for different values of c4 = {0.3, 0.4, . . . , 0.9, 1} and c5 = {0.1, 0.5, 0.7, 1} when η = 1, E = 1, and σ0 = 1. The charts also show that using this monotonically increasing VO-function generates unrealistic oscillations in the primary creep. Moreover, setting α(t) = c4 (1 − 1+c1 t 2 ) results 5 in a similar unrealistic response. As a result, it is shown that the VO-functions for the proposed rheological models should be monotonically decreasing to obtain realistic creep responses.

6 Numerical examples In this section, the constitutive equation (29) with monotonically decreasing VOfunctions is used to estimate the creep behavior of the 3M™ Scotch-Weld™ EC-2216 B\A Gray (EC-2216). The EC-2216 is a popular epoxy that is mainly used in aerospace and optomechanical industries. The creep tests are obtained by uniaxial tensile tests of the single lap joint, which is a common bonded joint in the industry. The method of collecting the creep data is described in [8], where the experiment was performed by an apparatus designed similar to (but not exactly the same as) the one proposed in the ASTM D2294. It was assumed that the creep law in the single lap joint is only dependent on the applied shear stress and the shear strain. In addition, the adhesive and adherents were assumed to be rigid layers such that the stress has a uniform distribution in the adhesive layer. Figure 8 illustrates three isothermal experimental creep data obtained from shear tests [8]. Two experimental data (S43 and SC5) are used to estimate the creep models, and the third experimental data (SF2) is used to verify the estimated models.

Figure 8: Three isothermal experimental creep data for the EC-2216 [8].

220 | B. Parsa Moghaddam et al. By considering the results of Section 5, the following monotonically decreasing VOfunction with two unknowns are considered for constructing the VOFDOs: α(t) = c4 exp(−c5 t), β(t) = c6 exp(−c7 t),

0 ≤ c4 ≤ 1, 0 ≤ c5 ≤ 10, 0 ≤ c6 ≤ 1, 0 ≤ c7 ≤ 10.

(31)

Next, an optimization method is used to find the optimal values of the unknowns ci , i = 1, 2, . . . , 7, by minimizing the squared 2-norm of the residual at the discretized 2 points goodness, that is, by minimizing ∑ni=1 (ŷi − yi ) , where yi and ŷi are the discretized strain of the model and experimental model, respectively. The optimization algorithm is based on discretizing equation (29) by using the SM-algorithm formulated in Propositions 1 and 2 with h = 0.005, so that the approximation error can be determined by using Proposition 3. Table 1: The minimized norm of residual squared (MNRS) and optimal parameters of the integerorder, the fixed-order fractional, and the ν1- and ν2-type VOF models with α(t) = c4 exp(−c5 t), β(t) = c6 exp(−c7 t), and t ≤ 1. Model Integer-order Fixed-order fractional ν1-type VOF ν2-type VOF

MNRS

c1

c2

c3

c4

c5

c6

c7

6

1.81 2.10 1.93 1.83

6.26 9.95 9.49 9.15

19.9 19.6 18.7 17.9

1 0.398 0.294 0.254

0 0 0.973 0.968

1 0.274 0.141 0.114

0 0 1.19 1.22

1.01×10 7.61×103 6.12×103 5.73×103

Table 1 shows the obtained values for the unknown of the integer-order (i. e., c4 = c6 = 1 and c5 = c7 = 0), the fixed-order fractional (i. e., c5 = c7 = 0), and the VOF constitutive equations for the loading regime when t ≤ 1. The results show that the ν2-type gives the best fitting of all proposed models. Moreover, there is a significant improvement in the creep estimation by using the fractional rheological models. Figure 9 shows the creep response and the variation of α(t) and β(t) of the proposed rheological models. It shows that α(t) of the ν1- and ν2-type models converges to 0 for t ≥ 1.

7 Conclusions It has been shown that constitutive equations with VOFDOs can successfully describe the behavior of viscoelastic materials and polymers. Recently, the use of VOFDOs has been the focus of attention in many different research fields. However, there has been much debate about how to choose the variable-order function in VOFDOs. This chapter reviewed several VOF rheological models for describing the creep process. It

Application of variable-order fractional calculus in solid mechanics | 221

Figure 9: The creep response and the variation of the orders α(t) and β(t) in the approximated VOF rheological models for the EC-2216 experimental creep tests. The line with the mark ∘ indicates the integer-order model; the line with the mark + indicates the fixed-order fractional model; the line with the mark ⋆ indicates the ν1-type VOF model; the line with the mark × indicates the ν2-type VOF model.

was shown that the functions associated with the variable orders should be chosen monotonically decreasing to obtain a typical creep process. Furthermore, experimental creep tests of an epoxy were used to estimate integer-, fixed-, and variable-order fractional rheological models. It was also shown that VOF rheological models are significantly better in estimating the creep curve in comparison with integer-order rheological models. Moreover, VOF rheological models with ν2-type VOFDOs are superior to those with fixed- and ν1-type VOFDOs.

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Jan Terpak

Fractional heat conduction models and their applications Abstract: This contribution deals with fractional heat conduction models and their applications. A brief historical overview of the authors who have dealt with the heat conduction equation is given in the introduction. The one-dimensional heat conduction models using integer- and fractional-order derivatives are listed. Numerical methods of solution of the heat conduction models using integer- and fractional-order derivatives for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions are described. In the case of numerical methods we deal with the finite-difference method using the Grünwald–Letnikov definition for the fractional time derivative. Implementation of these individual methods was realized in Matlab. A library of m-functions for the fractional heat conduction model has been created, namely the Time Fractional-Order Diffusion-Wave Equation Toolbox. The simulation examples using this toolbox are listed. At the end of the contribution applications are presented such as experimental verification of the methods for determining thermal diffusivity using the half-order derivative of the temperature by time. Keywords: Fractional heat conduction model, partial differential equation, Grünwald– Letnikov derivative, Matlab toolbox, simulation, thermal diffusivity MSC 2010: 35-02, 65-02, 80A20, 35K05, 65C20, 65N06, 68U20

1 Introduction The heat conduction process, described by partial differential equations, was first formulated by Jean Baptiste Joseph Fourier (1768–1830). In 1807 he wrote an article called Partial differential equation for heat conduction in solids. The issue of heat conduction was addressed by other scientists as well, such as Fick, Maxwell, Einstein, Richards, and Taylor [33]. Various analytical and numerical methods are used for the solution of the Fourier heat conduction equation [15, 38]. In the case of heat conduction in materials with non-standard structure, such as polymers, granular and porous materials, and composite materials, a standard description is insufficient and the creation of more adequate models is required by using derivatives of fractional order [21, 45, 46, 53]. The causes are mainly the memory of the systems and of the running processes [42, 43, 49, Jan Terpak, Technical University of Kosice, BERG Faculty, B. Nemcovej 3, 04200 Kosice, Slovakia, e-mail: [email protected] https://doi.org/10.1515/9783110571905-012

226 | J. Terpak 50, 58], roughness or porosity of the material [35–37], and also fractality and chaotic behavior of systems [6, 30, 39–41]. The subject of research and development methods and tools for process modeling by using fractional-order derivatives is hot, since it means a qualitatively new level of modeling. Important authors of the first articles were Fourier, Abel, Leibniz, Grünwald, and Letnikov. Mathematicians like Liouville (1809–1882) [23, 24] and Riemann (1826–1866) [47] made major contributions to the theory of fractional calculus. Nowadays there are a number of analytical [2, 7, 12, 16, 25, 29, 51, 60] and numerical solutions of fractional heat conduction equation. In the case of numerical methods different methods are being developed based on the random walk models [13, 14, 27, 26], the finite-difference method [52, 55, 64], the finite-element method [8–10, 48], numerical quadrature [1, 19, 62], the method of Adomian decomposition [18, 32], Monte Carlo simulation [11, 31], matrix approach [44–46], or the matrix transform method [17, 61]. The finite-difference method is an extended method where an explicit [22, 52, 63], an implicit [5, 20, 28, 64], and a Crank–Nicolson scheme are being used [55, 56]. For the Crank–Nicolson scheme, the literature describes the use of the Grünwald–Letnikov definition only for a spatial derivative [4, 5, 54, 59]. The work presented in this contribution is mainly aimed at the implementation of the finite-difference method for the time fractional-order diffusion-wave equation in Matlab and in the case of the Crank–Nicolson scheme applies the Grünwald–Letnikov definition for the time derivative.

2 Fractional heat conduction models Heat conduction is a molecular transfer of thermal energy in solids, liquids, and gases due to a temperature difference. The process of heat conduction takes place between the particles of the substances that touch each other directly and have different temperatures. Existing models of heat conduction processes are divided according to various criteria. We consider a division into two groups to models using derivatives of integer and fractional order. Models using derivatives of integer order are divided into non-stationary and stationary models. Non-stationary models are described by the Fourier heat conduction equation, where the temperature T (K) is a function of spatial coordinate x (m) and time τ (s). In the case of one-dimensional heat conduction it has the following form [38]: 𝜕T(x, τ) 𝜕2 T(x, τ) , = (√a)2 𝜕τ 𝜕x 2

(1)

where a = λ/(ρcp ) is thermal diffusivity (m2 ⋅ s−1 ), ρ is density (kg ⋅ m−3 ), cp is specific heat capacity (J ⋅ kg−1 ⋅ K−1 ), and λ is thermal conductivity (W ⋅ m−1 ⋅ K−1 ).

Fractional heat conduction models and their applications | 227

A heat conduction model using derivatives of fractional order for various onedimensional geometric cases was expressed by the following Oldham–Spanier equation [36]: 𝜕T(x, τ) 1 𝜕1/2 [T(x, τ) − T0 ] g[T(x, τ) − T0 ] + = 0, + √a 𝜕x x+R 𝜕τ1/2

(2)

where g is a geometric factor and R is a radius of curvature (m). In the case of one-dimensional heat conduction planar wall (g = 0) equation (2) shall take the following form: 𝜕1/2 [T(x, τ) − T0 ] 𝜕T(x, τ) ⋅ = −√a 𝜕x 𝜕τ1/2

(3)

A more general formulation of the task for modeling not only one-dimensional heat conduction is based on the model in which on the left-hand side of equation (1) instead of the first derivative with respect to time, the derivative of order α occurs, i. e., we can find it in the form [2] 2 𝜕α T(x, τ) 2 𝜕 T(x, τ) √ , a) = ( 𝜕τα 𝜕x 2

(4)

where √a represents a constant coefficient with the unit m ⋅ s−α/2 . Fractional-order models can also be described by the following equation (5), where α and β are of arbitrary order [44–46]: 𝜕β T(x, τ) 𝜕α T(x, τ) ⋅ = (√a)2 α 𝜕τ 𝜕|x|β

(5)

3 Numerical methods In this section, we derive the finite-difference methods for time fractional-order diffusion-wave equations (4) in one space dimension for inhomogeneous material (Figure 1).

Figure 1: Inhomogeneous rod.

228 | J. Terpak The fractional derivative in equation (4) is discretized by the backward Euler method and the Grünwald–Letnikov definition N(f ) 𝜕α T(x, τ) ∑j=0 bj Ti,p−j ≈ 𝜕τα Δτα

(6)

by using the principle of “short memory”, where τL is the “length memory” [43], Δτ is the time step, and the value of N(f ) will be determined by the following relation: N(f ) = min{[

τ τ ], [ L ]}⋅ Δτ Δτ

(7)

For the calculation of the binomial coefficients bj we can use the relation b0 = 1, bj = (1 −

1+α ) ⋅ bj−1 , j

for j ≥ 1.

(8)

Finite-difference methods according to the type of differential expression can be divided into explicit, implicit, and Crank–Nicolson schemes.

3.1 Explicit scheme Explicit scheme (Figure 2) for the time fractional-order diffusion-wave equation (4) has the form N(f ) bj Ti,p−j ∑j=0

Δτα

=

ai Ti+1,p−1 − (ai + ai−1 )Ti,p−1 + ai−1 Ti−1,p−1 Δx 2

.

(9)

The above equation is equivalent to the following form: N(f )

Ti,p = Mi−1 Ti−1,p−1 − (Mi−1 + Mi )Ti,p−1 + Mi Ti+1,p−1 − ∑ bj Ti,p−j , j=1

(10)

where module Mi is Mi = (

2

√ai ) Δτα , Δx

Figure 2: Stencil for explicit scheme.

(11)

Fractional heat conduction models and their applications | 229

while the value of Mi with respect to the stability of the solution must be less than or equal to 0.5. The matrix form of equation (10) is as follows: Tp = A . Tp−1 + B

(12)

or T2,p M1 [ .. ] ] [ . ]=[ [Tn−1,p ] [ [ [ [

−(M1 + M2 ) .. .

M2 .. . Mn−2

..

. −(Mn−2 + Mn−1 )

T1,p−1 N(f ) [ ] − ∑j=1 bj T2,p−j [ T2,p−1 ] [ ] [ ] [ .. ] [ ] . .. .[ . ]+[ ]. [ ] [ ] [T ] N(f ) − ∑j=1 bj Tn−1,p−j [ n−1,p−1 ] [ ] [ Tn,p−1 ]

] ] ]

Mn−1 ]

(13)

Solution of the time fractional-order diffusion-wave equation (4) requires two boundary conditions for spatial dimension, as well as one initial condition for the non-steady-state problem. The initial condition specifies the temperature distribution in the medium at the origin of the time coordinate T(x, 0) = f (x),

(14)

in discrete form Ti,1 = fi

for i = 1, . . . , n.

(15)

The boundary conditions specify the temperature or heat flux at the boundaries of the region. We shall consider the following three types of boundary conditions: 1. Dirichlet boundary conditions or boundary conditions of the first type – the temperature is prescribed at the boundary surface, that is, 󵄨 T(x, τ)󵄨󵄨󵄨x=0 = T0

󵄨 and T(x, τ)󵄨󵄨󵄨x=L = TL ,

(16)

and Tn,p = TL ,

(17)

and in discrete form T1,p = T0 2.

where T0 , TL is the temperature (K) at the boundary surface. Neumann boundary conditions or boundary conditions of the second type – the heat flux is prescribed at the boundary surface, that is, − λ(x)

𝜕T(x, τ) 󵄨󵄨󵄨󵄨 =i 󵄨 𝜕x 󵄨󵄨󵄨x=0 0

and

− λ(x)

𝜕T(x, τ) 󵄨󵄨󵄨󵄨 =i , 󵄨 𝜕x 󵄨󵄨󵄨x=L L

(18)

230 | J. Terpak and in discrete form T1,p = T2,p + i0 R0

3.

and Tn,p = Tn−1,p + iL RL ,

(19)

where i0 , iL is heat flux (W ⋅ m−2 ) and R0 , RL is internal (Δx/λ) thermal resistance (m2 ⋅ K ⋅ W−1 ) at the boundary surface. Robin boundary conditions or boundary conditions of the third type – the heat convection is prescribed at the boundary surface, that is, − λ(x)

𝜕T(x, τ) 󵄨󵄨󵄨󵄨 = h0 (T|x=0 − Ts0 ) 󵄨 𝜕x 󵄨󵄨󵄨x=0

(20)

− λ(x)

𝜕T(x, τ) 󵄨󵄨󵄨󵄨 = hL (T|x=L − TsL ), 󵄨 𝜕x 󵄨󵄨󵄨x=L

(21)

r 1 T + 0 T 1 + r0 2,p 1 + r0 s0

(22)

r 1 Tn−1,p + L TsL , 1 + rL 1 + rL

(23)

and

and in discrete form T1,p = and Tn,p =

where h0 , hL is the heat transfer coefficient (W ⋅ m−2 ⋅ K−1 ); Ts0 , TsL is surroundings temperature (K); and r0 , rL is the ratio of internal (Δx/λ) and external (1/h) thermal resistance at the boundary surface.

3.2 Implicit scheme The implicit scheme (Figure 3) for the time fractional-order diffusion-wave equation (4) has the form N(f ) bj Ti,p−j ∑j=0

Δτα

=

ai Ti+1,p − (ai + ai−1 )Ti,p + ai−1 Ti−1,p Δx 2

Figure 3: Stencil for implicit scheme.

(24)

Fractional heat conduction models and their applications | 231

or N(f )

− Mi−1 Ti−1,p + (1 + Mi−1 + Mi )Ti,p − Mi Ti+1,p = − ∑ bj Ti,p−j . j=1

(25)

This scheme gives a tridiagonal system of equations to solve for all the values Ti,p . In matrix form this is A . Tp = B

(26)

or A1,1 [ [ −M 2 [ [ [ [ [ [ [ [

−M2

(1 + M2 + M3 ) .. . −Mn−3

−M3 .. . (1 + Mn−3 + Mn−2 ) −Mn−2

..

. −Mn−2 An−2,n−2

B1,1 ] [ N(f [ − ∑ ) bj T3,p−j ] j=1 ] [ ] [ .. ]. =[ . ] [ ] [ N(f ) ] [− ∑ b T j n−2,p−j ] [ j=1 Bn−2,1 ] [

] [ T2,p ] ] [ T ] ] [ 3,p ] ] [ . ] ].[ . ] ] [ . ] ] [ ] [Tn−2,p ] ] ] T n−1,p ] ] [

(27)

The selected elements of matrix A and B (equation (27)) are given by the boundary conditions listed in Table 1. Table 1: Selected elements of matrix A and B for implicit scheme. Selected elements A1,1 =

An−2,n−2 = B1,1 =

Bn−2,1 =

1 + M2 +

1 + Mn−2 +

N(f ) − ∑j=1 bj T2,p−j N(f ) − ∑j=1 bj Tn−1,p−j

Boundary conditions Dirichlet Neumann

Robin

M1

M1 1+r0

0

Mn−1

0

+

M1 T1,p

M1 i0 R 0

+

Mn−1 Tn,p

Mn−1 iL RL

r

0

r

Mn−1 1+rL

L

r M1 1+r0 Ts0 0 r Mn−1 1+rL TsL L

3.3 Crank–Nicolson scheme The Crank–Nicolson scheme (Figure 4) for the time fractional-order diffusion-wave equation (4) has the form

232 | J. Terpak

Figure 4: Stencil for Crank–Nicolson scheme. N(f ) bj Ti,p−j ∑j=0

Δτα

=

ai Ti+1,p−1 − (ai + ai−1 )Ti,p−1 + ai−1 Ti−1,p−1 +

2Δx2 ai Ti+1,p − (ai + ai−1 )Ti,p + ai−1 Ti−1,p 2Δx2

or

1 (−Mi−1 Ti−1,p + (2 + Mi−1 + Mi )Ti,p − Mi Ti+1,p ) 2

N(f )

1 = (Mi−1 Ti−1,p−1 − (Mi−1 + Mi )Ti,p−1 + Mi Ti+1,p−1 ) − ∑ bj Ti,p−j . 2 j=1

(28)

(29)

In matrix form this is 1 1 A .T = A .T +B 2 p p 2 p−1 p−1

or A1,1 [ [ −M 2 [ 1[ [ 2[ [ [ [ [

M1 1[ = [ 2[ [

(30)

−M2

(2 + M2 + M3 ) .. . −Mn−3 −(M1 + M2 ) .. .

−M3 .. . (2 + Mn−3 + Mn−2 ) −Mn−2 M2 .. . Mn−2

B1,1 ] [ [ − ∑N(f ) bj T3,p−j ] j=1 ] [ ] [ .. ]. +[ . ] [ ] [ N(f ) ] [− ∑ b T j n−2,p−j ] [ j=1 Bn−2,1 ] [

..

..

.

−Mn−2 An−2,n−2

. −(Mn−2 + Mn−1 )

] [ T2,p ] ] [ T ] ] [ 3,p ] ] [ . ] ].[ . ] ] [ . ] ] [ ] [Tn−2,p ] ] ] T ] [ n−1,p ]

T1,p−1 ] [ . ] ].[ . ] ] [ . ] Mn−1 ] [Tn,p−1 ]

(31)

The selected elements of matrix Ap and B (equation (31)) are given by the boundary conditions listed in Table 2.

Fractional heat conduction models and their applications | 233 Table 2: Selected elements of matrix Ap and B for Crank–Nicolson scheme. Selected elements A1,1 =

An−2,n−2 = B1,1 =

Bn−2,1 =

2 + M2 +

2 + Mn−2 +

N(f )

− ∑j=1 bj T2,p−j + N(f )

− ∑j=1 bj Tn−1,p−j +

Boundary conditions Dirichlet Neumann

Robin

M1

M1 1+r0

Mn−1

M1 T 2 1,p Mn−1 Tn,p 2

0 0

M1 i R 2 0 0 Mn−1 i R 2 L L

r

0

r

Mn−1 1+rL

L

M1 r 0 T 2 1+r0 s0 Mn−1 rL T 2 1+rL sL

4 Implementation The implementation was realized in the programming environment Matlab in which the functions for solution of the time fractional-order diffusion-wave equation have been created. A library of m-functions called Time Fractional-Order Diffusion-Wave Equation Toolbox (TFODWET) consists of three functions for solving one-dimensional models. Functions include numerical methods for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions (Section 3). The shape of the declaration functions are as follows: function [Xco,Tau,MU] = TFODWE_explicit (Uin,a,Nf,b,dx,L,dt,T,TBC,BC) function [Xco,Tau,MU] = TFODWE_implicit (Uin,a,Nf,b,dx,L,dt,T,TBC,BC) function [Xco,Tau,MU] = TFODWE_CrankNicolson (Uin,a,Nf,b,dx,L,dt,T,TBC,BC)

where function inputs are Uin initial condition, a derivative order, Nf length memory, b coefficients, dx spatial step, L distance, dt time step, T time of simulation, TBC type of boundary conditions, and BC boundary conditions, and outputs are Xco vector of x-coordinate, Tau vector of time, and MU matrix of output values. The toolbox TFODWET is published at Mathworks, Inc., Matlab Central File Exchange [57].

5 Simulations Simulations for homogeneous or inhomogeneous material can be implemented by using the script TFODWE_test.m, which is part of the library TFODWET. The initial conditions, thermophysical properties of the material, spatial division and thickness of the material, time step and total simulation time, type and values of the boundary conditions, and method of solving are defined in the given script. Based on the above definitions the simulation, which is the result of behaviors of temperatures in space and time, can be implemented. As an illustration we give an example of a simulation, where the initial conditions have been defined in the form of the

234 | J. Terpak

Figure 5: The course of temperatures in time and space for homogeneous material.

Figure 6: Three-dimensional display of temperatures in time and space for homogeneous (a) and inhomogeneous (b) material.

same temperatures of 20 °C after cross-section, copper was considered (thermal diffusivity = 1.1236 × 10−4 m2 ⋅ s−1 ), material thickness was 0.02 m, the material was divided into twenty parts in the space, the time step was 0.001 s, the total time simulation was 2 s, under Dirichlet boundary conditions with temperatures at the edges T0 = 0 °C and TL = 25 °C, and the Crank–Nicolson scheme was chosen. Simulation results are shown in Figure 5 and Figure 6a. Shown is the behavior of temperature in time, where the parameter is the position in space, then it is behavior of temperature in space, where the parameter is time (Figure 5), and finally we show a three-dimensional display of temperatures in space and time (Figure 6a). The script TFODWE_test.m is also intended to simulate inhomogeneous material. Due to the larger number of materials the individual properties are given as a vector, i. e., each component of the vector represents the thermal diffusivity of a given layer. For example, in the case of thermal diffusivity for twenty layers and two materials (brass and copper) of the same thickness a twenty-component thermal diffusivity vector is needed.

Fractional heat conduction models and their applications | 235

Figure 7: Three-dimensional display of temperatures in time and space for Dirichlet boundary conditions.

Figure 8: Three-dimensional display of temperatures in time and space for Neumann boundary conditions.

Results of the simulations are shown in Figure 6b, where in the first half of the length there is a lower temperature rise than in the other half. This is due to the fact that in the first half there is copper, whose thermal conductivity is about three times larger than that of brass. The following simulations demonstrate the use of different homogeneous boundary conditions for homogeneous material. The behavior of temperatures for the Dirichlet (Figure 7), Neumann (Figure 8), and Robin (Figure 9) boundary conditions are shown. In the case of the Dirichlet boundary conditions, the surface temperature is given (T0 = TL = 25 °C); in the case of the Neumann boundary conditions, the thermal flux and the internal thermal resistance on the surface are given (i0 ⋅ R0 = iL ⋅ RL = 0.8 °C), and in the case of the Robin boundary conditions the surroundings temperature and the ratio of internal and external thermal resistance are given (Ts0 = TsL = 25 °C, r0 = rL = 1). In Figure 10–12 inhomogeneous boundary conditions are shown for selected combinations, i. e., Neumann–Dirichlet, Robin–Dirichlet, and Neumann– Robin.

236 | J. Terpak

Figure 9: Three-dimensional display of temperatures in time and space for Robin boundary conditions.

Figure 10: Three-dimensional display of temperatures in time and space for Neumann and Dirichlet boundary conditions.

Figure 11: Three-dimensional display of temperatures in time and space for Robin and Dirichlet boundary conditions.

Fractional heat conduction models and their applications | 237

Figure 12: Three-dimensional display of temperatures in time and space for Neumann and Robin boundary conditions.

Figure 13: Three-dimensional display of temperatures in time and space for α = 0.5.

Figure 14: Three-dimensional display of temperatures in time and space for α = 1.0.

238 | J. Terpak

Figure 15: Three-dimensional display of temperatures in time and space for α = 1.5.

Figure 13–15 show the evolution results for α = 0.5, 1.0, and 1.5, respectively. Comparison of Figure 13 and 14 shows that the half-order derivative exhibits a fast temperature rise in the beginning and a slow temperature rise later. Figure 15 shows temperatures propagate and diffuse with time, which means that temperatures continuously depend on the fractional derivative. Therefore, when α = 1.5, both diffusion and a wave response can be observed.

6 Applications The experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time is provided in the next subsections.

6.1 Thermal diffusivity determination method The method is derived from the calculation of heat flows. Determination of the heat flow iQ is possible in two ways (equation (3)), namely: 1. from the gradient of the two measured temperatures, i. e., iQ = −λ 2.

𝜕T(x, τ) ; 𝜕x

(32)

from the half-order derivative of one measured temperature, i. e., iQ =

λ 𝜕1/2 [T(x, τ) − T0 ] . √a 𝜕τ1/2

(33)

Fractional heat conduction models and their applications | 239

Figure 16: Measured temperatures.

The share of the half-order derivative and the gradient of the temperature is proportional to the square root of the thermal diffusivity, i. e., √a =

𝜕1/2 [T(x, τ) − T0 ] 𝜕τ1/2 ⋅ 𝜕 T(x, τ) − 𝜕x

(34)

The differential form of equation (34) is shown in the form √a =

N(f ) Δτ−1/2 ∑j=0 bj [T1,p−j − T0 ]

Δx−1 [T2,p − T1,p ]

⋅

(35)

For the numerical calculation of the first derivative of temperature according to the coordinate, respectively, temperature gradient (34), it is sufficient to measure two temperatures (Figure 16). The calculation of thermal diffusivity is based on the ratio of the half-order derivative of temperature according to time to the temperature gradient (Figure 17), which is observed based on the values of two neighboring temperatures in space obtained from simulations. More previous values of temperatures in time are used for the calculation of the half-order derivative, as in the case of the first derivative, which uses only one previous value. The method was tested on the model using the Crank–Nicolson scheme on a brass sample. The initial temperature of simulation was set at 20 °C, with Dirichlet boundary conditions of 20 °C and 100 °C and with a time step of the simulation 0.01 s. Input

Figure 17: Temperatures and ratio of the half-order derivative to the gradient of the temperature.

240 | J. Terpak

Figure 18: Crank–Nicolson scheme for a time step of 0.1 s and 0.05 s.

parameters of the brass were the following: density 8 400 kg⋅m−3 , specific heat capacity 380 J ⋅ kg−1 ⋅ K−1 , and thermal conductivity 120 W ⋅ m−1 ⋅ K−1 . The value of thermal diffusivity for brass is a = 3.4130 × 10−5 m2 ⋅ s−1 , or √a = 0.0061314 m ⋅ s−1/2 . In Figure 18 we can see the effect of the time step to calculate the square root of thermal diffusivity. The calculation accuracy of determining the value of the square root of thermal diffusivity depends on the number of previous values of temperatures in time and also on the selected time step. Reducing the number of previous values of temperatures leads to a higher inaccuracy of the calculation.

6.2 Method verification The method has been verified on the experimental measurements. Measurements were carried out on the devices HT10XC and HT11C. Module HT11C is a physical model of one-dimensional heat conduction [3]. It consists of a heating and cooling section, while the sample material is inserted between them (Figure 19).

Figure 19: Scheme of HT11C.

Fractional heat conduction models and their applications | 241

The brass sample was used in the form of a cylinder with a diameter of 25 mm and a height of 30 mm. Contact areas of the sample were coated with a thin layer of thermal paste to minimize the transient thermal resistance. Module HT11C uses thermocouples of type K in the temperature range from 0 to 133 °C and the distance among them is 15 mm. The device HT10XC with the HT11C module is connected via USB to a PC. The software that comes with the device allows to set conditions of the experiment and the measurement data can be saved to a file. Experimental measurements which are referred to in this section were carried out under the following conditions: heater power 1.3 W, the water flow in the cooler 0.5 L ⋅ min−1 , and the time step for recording measured data 1.0 s. A unit jump in the heater power from 1.3 to 3.3 W was realized after stabilizing the temperatures. The transition from one steady state to another is shown in Figure 20.

Figure 20: Experimental measurements of temperatures for the brass.

In Figure 21 the square root of thermal diffusivity is determined from the measured values of the device HT11C, that is, from the ratio of the half-order derivative of the temperature according to time to the temperature gradient. The value of thermal diffusivity of the used brass sample for equipment HT11C was 3.2233 × 10−5 m2 ⋅ s−1 and this corresponds to the square root of a thermal diffusivity of 0.0056774 m ⋅ s−1/2 . The brass sample was also measured on the device LFA [34] and the value of thermal diffusivity was 3.4130 × 10−5 m2 ⋅ s−1 , which corresponds to the square root of a thermal diffusivity of 0.0058421 m ⋅ s−1/2 . The calculated relative error between the measured values of the thermal diffusivity of the brass sample on HT11C and LFA is 5.5591 %.

242 | J. Terpak

Figure 21: Thermal diffusivity.

7 Conclusion The models of the heat conduction by using integer and fractional derivatives are listed and described in this chapter. They have been implemented in Matlab as library functions, i. e., the Time Fractional-Order Diffusion-Wave Equation Toolbox (TFODWET) [57]. Possible ways to use this library are illustrated in the examples. Besides modeling the processes of heat conduction the library also allows to model the processes of diffusion, wave, etc. The library is an appropriate tool for the creation of complex models in Matlab. Benefits of this tool include the use of the Grünwald–Letnikov definition of the time derivative in the case of the Crank–Nicolson scheme for inhomogeneous material and for inhomogeneous boundary conditions. The tool includes methods to design, implement, and verify thermal diffusivity models using the half-order derivative of temperature according to time and experimental equipment.

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Index acute pain 56 analgesia 56, 65 anomalous diffusion 60, 69, 91 anomalous transport 193 anti-resorptive therapy for bone tumor 73 augmented multi-model 114, 115 auxetic material 151 basic multi-cellular unit (BMU) 71 binomical coefficients 228 bio-electrical impedance 58, 65 bio-inspired robotics 95 bisphosphonates 73 Boltzmann–Volterra relationships 163 bone remodeling 70 bone tumor 72 Boussinesq equation 121 bulk relaxation 155 Caputo fractional derivative 3 central nervous system 57 chemotherapy 73 chronic pain 56 Cole–Cole 24 Cole–Davidson 24 creep 207, 212 creep compliance 212 Darcy’s equation 121 Debye 23, 24 defect of the modulus 143 dimensionless Rabotnov fractional operator 172, 176 Dirichlet boundary conditions 229 drainpipes 125 Electrical impedance spectroscopy 21 electrode 62, 65 equilibrium points 4, 14 existence and uniqueness 12 Fick’s law 193 Finite-difference method – Crank–Nicolson scheme 231 – explicit scheme 228 – implicit scheme 230 Fourier heat conduction equation 226 https://doi.org/10.1515/9783110571905-013

fractional calculus 1 fractional calculus viscoelasticity 159 fractional derivative 193 fractional derivative Kelvin–Voigt model 140, 141, 149, 150, 169 fractional derivative Maxwell model 140, 142, 151 fractional derivative standard linear solid model 140, 143 fractional governing equation 122 fractional multi-model 102 fractional order derivatives 60 fractional order HIV model 3 fractional order impedance models 59 fractional order tumor model 12 fractional oscillator 163 fractional-order models 43 frequency of vibrations 164 frequency response 64 generalized Hertzian contact law 173, 177 generalized Rabotnov model 147 Gerasimov–Caputo derivative 140 groundwater 120 Grünwald–Letnikov approximation 6 Grünwald–Letnikov derivative 228 Havriliak–Negami 24 indentation 177, 179 instantaneous compliance 142 inversion Mellin–Fourier formula 164 isometric condition 96, 115 isometric contraction 95, 96, 98, 100, 115 isometric responses 99, 107, 114 isotonic 96 isotonic contraction 96, 98 Jacobian matrix 5 Jordan lemma 164 Kelvin–Voigt 213 Kirchhoff–Love plate 171 Laplace transformation 164, 175 linear B-spline 210

248 | Index

mathematical drainage model 130 Matlab 233 Maxwell 213, 214 Mellin–Fourier inversion formula 175 Mittag-Leffler function 60 monoclonal antibodies 73 motor neuron 95–98 motor unit 95–104, 106–113, 115 multi-model 95–97, 100–102, 106–111, 114, 115 multi-model system identification 101 multi-scale processes 56 multiple mieloma (MM) 73 multisine 62 muscle 95 negative Poisson’s ratio 151 Neumann boundary conditions 229 nociceptor pathways 55 nociceptor stimulation 64, 65 nociceptors 57 non-integer model 95, 96, 100 non-integer modeling 95 non-integer multi-model 95–97, 108, 115 non-parametric identification 62 non-relaxed compliance 164 non-relaxed (instantaneous, or glassy) modulus of elasticity 142 non-relaxed modulus 164 non-relaxed shear modulus 151 numeric rating scale 56 numerical simulations 17 operator of rigidity 178 osteoblasts 71 osteoclasts 71 pain 55, 63 pain memory 62, 64, 65 parametric parsimony 95, 96 periodogram 63 pharmacokinectics and pharmacodynamics (PK/PD) 73–75 Poisson’s operator 178 Rabotnov fractional exponential function 142, 167

Rabotnov fractional operator 141 Rayleigh hypothesis 171, 172 relaxation 207, 212 relaxation kernel 163, 167 relaxation modulus 213 relaxation time 142, 146 relaxed comliance 164 relaxed modulus 164 relaxed shear modulus 150 reproducing kernel Hilbert space 44 resolvent operators 143, 145 retardation kernel 163 retardation time 146, 149 rheological 215 Riemann–Liouville derivative 140, 145, 149, 169 Riemann–Liouville fractional integral 3 Riemann–Liouville integral 140, 141 Robin boundary conditions 230 sampling frequency 62, 63 Scott Blair element 153 shear operator 151 singular time-fractional partial differential equations 43 skeletal muscle 95–98 skin impedance 55 stability analysis 4, 15 system identification 95, 96, 100, 105–108 time fractional-order diffusion-wave equation 227 tumor model 11 Uflyand–Mindlin plate 173 variable-order derivatives 70, 83, 89, 91 variable-order fractional derivative 207 variable-order function 209 viscoelastic Bernoulli–Euler beam 169, 176 viscoelasticity 212 volumetric relaxation 154 Young’s operator 149, 150, 169, 178 Zener 213, 214