Proceedings of the Fifth International Colloquium on Differential Equations: Plovdiv, Bulgaria, 18–23 August, 1994 [Reprint 2020 ed.] 9783112314029, 9783112302750

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Proceedings of the Fifth International Colloquium on Differential Equations

Proceedings of the

Fifth International Colloquium on Differential Equations Plovdiv, Bulgaria, 18 - 2 3 August, 1994

Editors: D. Bainov and V. Covachev

///VSP/// Utrecht, The Netherlands, 1995

VSP B V P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1995 First p u b l i s h e d in 1 9 9 5 ISBN 90-6764-192-8

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Koninklijke

Wöhrmann bv,

Zutphen.

CONTENTS Preface

1

Inverse Scattering Problem for the Wave Equation with Discontinuous Wavespeed T. Aktosun, M. Klaus and C. van der Mee

3

Difference Methods for Impulsive Partial Differential Equations D. Bainov, Z. Kamont, E. Minchev and K. Paczkowska-Przadka

23

On Radially Symmetric Solutions of Semilinear Elliptic Equations J. V. Baxley

39

Some Weighted Poincare-type Inequalities in One and Higher Dimensions R.C. Brown, D.E. Edmunds, D.B. Hinton and J. Rdkosnik

49

Matrix Nonstationary Semigroups: Semigroup Property and Relativity J.-P. Caubet

61

Critical Points Theory and its Application to a Nonlinear Wave Equation Q.-H. Choi and T. Jung

73

Existence of Decreasing Positive Solutions of a System of Linear Differential Equations with Delay J. Diblik

83

Equations of Nonlinear Membrane Theory R. W. Dickey

95

Analysis of Dynamics of Clustering Neural Networks L.V. Fausett

103

Elliptic Equations with Measure Data T. Gallouet A New B-Spline Finite Element Method for the Advection-Diffusion Equation L.R.T. Gardner and G.A. Gardner Existence of a Solution to a Coupled Elliptic System Arising in the Mathematical Modelling of a Fuel Cells R. Herbin

113 123

133

Comparing the Deficiency Indices of Powers of T+T and TT + , Where r is a Linear Ordinary Differential Expression F.D. Isaacs

143

Reproducing Kernels for Infinite Dimensional Domains J. Kajiwara and L. Li

153

Transform Inverse Formula J. Kajiwara and for M. Laplace Tsuji

163

vi

contents

Hankel Transform to the Diffusion of Dust Problem S.L. Kalla and E. Urribarri

173

Asymptotic D. Kannan Volume Nullification of Singular Diffusions

181

Conditions for the Absence of Positive Solutions of a First Order Differential Inequality with Distributed Delay E. Kozakiewicz

187

Stability X. Liu of Large Scale Impulsive Systems

193

The Heat Conduction in a Medium Enclosed by a Thin Shell of Higher Diffusivity Y. Luo

205

Exact Radial Asymptotics of Solutions to Singular Elliptic Differential Equations G. Lysik and B. Ziemian

213

Boundary Value Problems for Nonlinear Non-Variational Elliptic and Parabolic Systems: Solvability and Regularity Results A. Maugeri

223

On a Perturbation of the Schrodinger Equation — Exact Controllability L.A. Medeiros and M.M. Miranda

235

Modern Wiener-Hopf Methods in Diffraction Theory E. Meister

245

Wavelet Transforms in R n — Wave Front Sets and Besov, Triebel-Lizorkin Spaces S. Moritoh

257

A Decomposition of the Solutions of the Differential Equations of the Form x + (p(i)||a;||2*-2a: = F(t)x S.I. Nenov

267

Dirichlet Problem for a Class of Second Order Nonlinear Elliptic Equations D.K. Palagachev

273

Nonexistence of Isolated Singularities for Semilinear Systems of Partial Differential Equations P.R. Popivanov

283

Practical Taylor Series L.B. Rail

287

contents

vii

The Elastic Contact Problem for Sliding Cylinders A. Sackfield and M. Keysell

297

Landau-Ginzburg Dynamics and Geometric Evolution Equations B.E. Stoth

303

The Interior Stress Field for Two and Three Dimensional Linear Indenters C.E. Truman, A. Sackfield and D.A. Hills

313

The Solution of a Boundary Value Problem in Elasticity C.E. Truman, A. Sackfield and D.A. Hills

323

Round-off Errors in Floating-Point Solutions in Newton's Method for Two Dimensional Linear Systems of Equations K. Tsuji

331

Pseudo in Banach T. Tsuji Inverse and Z.-J. Yang Space for Newton Method

341

Present Status on the Method of Generalized Quasilinearization A.S. Vatsala

345

Recent Developments in Collocation Methods for Ordinary Differential Equations K. Wright

353

Preface The Fifth International Colloquium on Differential Equations was organized by the Mathematical Society of Japan, the Department of Mathematics of the Catania University, and UNESCO with the cooperation of the Association Suisse d'Informatique, the Canadian Mathematical Society, Hamburg University, Kyushu University, the London Mathematical Society, Plovdiv University "Paissii Hilendarski", Technical University — Berlin, Technical University — Plovdiv, Union of the Bulgarian Mathematicians — Plovdiv Section, the University of Texas — Pan American, and sponsored by the foundation "Evrika" and the "Pharmaceutical Scientific and Production Enterprise". It was held on August 18-23, 1994, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 18 to 23 of August in Plovdiv, Bulgaria. The address of the Organizing Committee is: Stoyan Zlatev, Mathematical Faculty of the Plovdiv University, Tsar Assen Str. 24, Plovdiv 4000, Bulgaria. The

Editors

5th Int. Coll. on Differential Equations, p. 3-22 D. Bainov and V. Covachev (Eds) © VSP 1995

INVERSE

SCATTERING

PROBLEM

F O R T H E WAVE

EQUATION

W I T H DISCONTINUOUS W A V E S P E E D

Tuncay Aktosun1, Martin Klaus2, and Cornelis van der Mee3 'Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA 3Department of Mathematics, University of Cagliari, Cagliari, Italy 2Department

Abstract: The inverse problem on the line is studied for the generalized Schrodinger equation (Pt^/dx2 + k2H(x)2ip = Q(x)%!>, where k is the wavenumber, 1/H(x) is the wavespeed, and Q(x) is the restoring force per unit length. H{x) is a positive, piecewise continuous function having limits H± as x —+ ±oo, and Q(x) satisfies certain integrability conditions. This equation describes wave propagation in a nonhomogeneous medium in which the wavespeed is allowed to change abruptly at certain interfaces. The inverse problem considered here consists in determining the function H(x) from a suitable set of scattering data and for a given Q(x). At the heart of the solution are a RiemannHilbert problem and a related singular integral equation. The solvability of the integral equation is discussed, and the solution method is illustrated by some explicitly solved examples. Keywords: 1-D Schrodinger equation, Inverse scattering, 1-D wave equation, Energydependent potential, Acoustics, Discontinuous wavespeed, Wave propagation, Singular integral equation

1. Introduction In this article we report on recent work concerning the inverse scattering problem for the one-dimensional generalized Schrodinger equation V>"(fc, x) -(- k2H(x)2ip(k, x) = Q(x)tjj(k, x),

IER,

(1.1)

where the prime denotes the ^-derivative. The functions H(x) and Q(x) obey certain conditions that will be detailed below. In the context of this article (1.1) describes

4

Aktosun, Klaus, van der Mee

the propagation of waves in a nonhomogeneous medium where k is the wavenumber, 1/H(x)

is the wavespeed, and Q(x) is the restoring force (per unit length). The function

H(x) may have j u m p discontinuities, that is, we allow for the possibility of the physical properties of the medium to change abruptly at certain interfaces. In the time domain (1.1) is equivalent to

where u(x,t)

is the wave amplitude and c(x) = 1/H(x)

is the wavespeed. Conversely,

(1.1) is the frequency domain version of (1.2). The conditions that H(x) and Q(x) need to satisfy are as follows: (HI)

H(x) is strictly positive and piecewise continuous with j u m p discontinuities at xn for n = 1, • • • ,N.

(H2)

H(x) —> H± as x —• ±oo, where H± are positive constants.

(H3)

H — H± £ L1 ( R ± ) , where R - = ( - o o , 0) and R + = (0, +oo).

(H4)

H'(x) is absolutely continuous on every interval (xn, x n + i ) and 2H" H — 3 (H')2 belongs to L\(x„,xn+i)

(H5)

for n = 0, • • • ,N, where x0 = —oo and x^+i = +oo.

Q € L j + a ( R ) for some a G [0,1], where

is the Banach space of complex-

valued measurable functions f ( x ) on I such that Jj dx (1 + |x|)^ | / ( x ) | < +oo. Under these conditions, (1.1) has two linearly independent solutions, so-called Jost solutions, satisfying the boundary conditions

fi(k,x)=\

(eikH+x+o( 1), 1 pikH_x,L(k) +

l2K*f Mk,x)=

1 i {Tr(k)e

e

e~iH+z

-ikH-x

-(-

z-+oo, itH_

(1.3) + ( )

W ) + Me«»+< Tr(k)

'

+ o(l) +o{l),

o(l),

' x^+oo x^+cc, X

( l 4 )

—> —oo.

Here Tr(k) and Ti(k) are the transmission coefficients from the right and from the left, respectively, and R(k) and L(k) are the reflection coefficients from the right and from the left, respectively. The scattering matrix is defined by m

Ti(k) L(k)

R(k) Tr(k)

The scattering matrix will not play a direct role in this article; we introduce it mainly as a convenient notational device. For certain negative values of k2 (1.1) may have a solution belonging to £ 2 ( R ) ; such values of k2 will be referred to as bound state energies

Inverse Scattering with Discontinuous Wavespeed

5

and we put k = in with K > 0. Our assumptions on H(x) and Q(x) guarantee that the number of bound states is finite. We will denote the number of bound states by Af. Associated with each eigenvalue — Kj (k = inj, j = 1, • • • , JV) is the norming constant Uj in (3.17). There are several inverse problems that can be studied in the context of (1.1). For example: 1. The classical inverse problem, where H(x) = 1 and one is asked to determine the function Q(x) from the scattering data consisting of either R(k) or L(k) for k € R , the eigenvalues —Kj and their norming constants uj, j = 1, • - • ,N. 2. The same as problem 1, but with a given H(x) from a suitable class of functions. 3. The problem where Q(x) is given and one is asked to determine H(x) from an appropriate set of scattering data. The first problem is well understood [1,2,3]. When Q € L ] ( R ) one has a complete characterization of the scattering data and there is a one-to-one correspondence between the scattering data and the potentials in Z}(R). The second problem was solved in [4] along the lines of problem 1. The aim of this article is to study the third problem. The case when H(x) is continuous was studied in [5]; the case when Q(x) = 0 and H(x) has jump discontinuities was considered by Grinberg [6,7]. Here we will consider the general case when Q(x) ^ 0 and H(x) has jump discontinuities. The present article is based on [8], where more details can be found. The main difference between the case Q(x) = 0 and Q(x) / 0 is that in the former we have |iZ(fc)| < 1 for k £ R , while in the latter we may have iZ(0) = —1. This difference makes the case Q(x) ^ 0 more difficult to study. It turns out that for problem 3 "essentially" the same set of scattering data as for problem 1 is appropriate. We say "essentially" because, as we will see in the examples, it may be necessary also to know either the value of H+ or H- in order to determine H(x) uniquely. So we may consider H+ or H— to be part of the scattering data. However, we also present an example (Example 2, Section 4), where H+ cannot be chosen freely, but is determined by Q(x) and R(k). This suggests that for problem 3 the characterization of the scattering data is more difficult than for problem 1 and needs to be investigated further. This article is organized as follows. In Section 2 we present some results concerning the asymptotic behavior of S(fc) as k —• 0 and k —> ±oo. These results are essential for Section 3, where we formulate a key Riemann-Hilbert problem and solve it by converting

6

Aktosun, Klaus, van der Mee

it into a singular integral equation. We also discuss the unique solvability of this integral equation. In Section 4 we consider three examples. Examples 1 and 2 have been worked out in detail in [8] a n d are included here for illustrative purposes. Example 3 is new and we give most of the details. Except in a few instances it is not possible to give detailed proofs in this article. So we will often refer the reader to [8] for more information.

2 . S mall-i; a n d Large-fc b e h a v i o r of

S(k)

In this section we determine the asymptotic behavior o f S ( f c ) as k —> 0 and k —• ±oo. We let C + denote the upper-half complex plane and C + = C + U R its closure. T h e transmission coefficients can be extended meromorphically to C + , and we will analyze their behavior as k —> 0 and k —• oo in C + . Let [/; +00).

When Q(x) = 0, which is the case considered in [6,7], we have fi(0, x) = / r ( 0 , x) = 1 and this corresponds to the exceptional case with 7 = 1.

8

Aktosun, Klaus, van der Mee The next theorem describes the small-A: behavior of S ( k ) .

Theorem 2.3

Assume H - H± € ^(R.*)

311(1

Q € L } + a ( R ) for some a € [0,1).

Then: (i) In the generic case p(k) =-1

t(k) = - l + o(\k\a), . fc —• 0 in R ,

+ o(\k\a),

fc

r(k) = ick + o(\k\1+a),

—•> 0 in C + ,

(2.10)

where (2.11)

[f,(o,xy,fr(o,x)Y (ii) In the exceptional case T{k)= 2

h*?+hI

& —> 0 in C+,

+

e { k ) =

Hlt

2

(2-13)

* -

+ Hl+°{lkn>

where 7 is the constant defined in (2.9). provided we replace the error terms by

(2.12)

0 i n R

>

Both (i) and (ii) remain valid for a = 1,

0(k).

We remark that (i) follows directly from (2.1), (2.2), and (2.4)-(2.6). The proof of (ii) is more involved [8]. Next we consider the large-fc behavior of S(fc). We use the fact that although H(x) is discontinuous at xj (j — 1, • • • , N), we can, on each interval (xj, Xj+\) (j = 0,••• , TV), perform a Liouville transformation of the form y = y(x)=

f Jo

dsH(s),

tj>(k,x) = ~7L=4>(.Ky). \/H{x)

Under this transformation, in each interval (xj, Xj+i), standard Schrodinger equation 00 in C + r„,n+i(fc,a:„+i - 0 )

1 r tl+ i ) „+2(Ä;,a; I , + i

+0)

an+1(l + o(l)) ßn+lC2iky"+1

(1 +

0(1))

/3 n + 1 e _ 2 i * 9 " + 1 ( 1 + o ( l ) ) « „ + , ( 1 +

0(1))

(2.19)

10

Aktosun, Klaus, van der Mee

where 1 H(xn i(k,x)

=

T i ( k ) f t ( k , x )

and

t!>r{k,x)

—p(k)e

2iky

r( k ) =

T

r

( k ) f

r

( k , x )

of (1.1)

satisfy ji p. jl u( k , x )

i\ TT , i( w k)

>r(M).

R ( k )

rru\~\ m ' T r ( k )

r./,1 p r t—U (-k,x)

k €

R,

(3.4)

and hence using (2.5), (2.6), and (3.1) in (3.4) we obtain (3.3). | Eq. (3.3) constitutes a Riemann-Hilbert problem for the vector function Z(k, y); however, it is not a standard Riemann-Hilbert problem because

Z(k,y)

does not converge

+

to a constant vector as k —• oo in C . Our goal is to recast (3.3) as an integral equation. For each fixed y £ R \ {j/i, • • • , yw}, from (3.3) we have r ( k ) Z

r

( k , y ) = Z , ( - k , y ) + p(k)e

2 i k

»Z,(k,y),

k € R,

(3.5)

12

Aktosun, Klaus, van der Mee t ( 0 ) Z r (0, y) = Z,( 0, y) + p(0) 7,(0, y).

(3.6)

F+(k, x, y) = — t = [r(*) Z r (A, y) - r ( 0 ) Z r (0, y)], k y/H(x)

(3.7)

Define

F

~(k>»)

=

L /irr k y/H(x)

w) -

y)] •

(3.8)

Using Z,(0,y) y/H(x)

_ fi(0,x)

Zr(0,y) _ f r ( 0 , x )

y/Hi'

y/H[x)

*jHl

'

from (3.5)-(3.8), for k £ R and y € R \ {yi, • • • , yjv} we obtain -p(t)e2">F,(-k,x,v)

F+(l,i,v) - F - ( M , s ) =

Since we are considering the case without bound states, for x € R \ {xj, • • • , xN} and y £ R \ {yi; " ' » !/n}, F±(k, x,y) have analytic extensions in A: to C1*1, and F±(k, x, y) —• 0 as k —> oo in C ± . The detailed justification for this conclusion is given in [8] (Theorems 4.4 and 5.2). The behavior of F±(k,x,y) H(x) - H± as x

at k = 0 depends on the decay of Q(x) and

±oo. If Q £ L\+a(R)

I1(R-±)>

with a € (0,1) and H(x) - H± £

then, by Theorem 2.1 and (3.1), we have Zt(k,y)

- Z,(0,y)

= o(\k\a),

Zr(k,y)

- Z r ( 0 , y ) = o(\k\a),

k -» 0 in C + .

Also T(jfe) - t ( 0 ) = o(|jfc|a) by (2.10) and (2.12). It follows that F±(k,x,y)

belong to

the Hardy spaces H ± ( R ) for 1 < p < 1/(1 — a); if a = 1 a similar argument shows that F±(k,x,y)

belong to H ± ( R ) for all p £ ( l , + o o ) . Recall that the Hardy spaces H ± ( R )

are the spaces of analytic functions f ( k ) on C ± for which s u p e > 0 f*™ dk\f(k

± ie)\p

is finite. Associated with these spaces are the projection operators II± which project L P ( R ) onto H5-(R) given by 1

t°°

tit

It is known that II± are bounded and complementary projections on Lp(R)

when 1


r 2t» J-0o

s-k^iO

s

- w m p . y/H+

- ¿ v - J .

Inverse Scattering with Discontinuous Wavespeed Hence F-(k,x,y)

13

obeys the singular integral equation

F (k x v) - J -J

/h~ = -jj-—Ji{0,x)

Defining F^(k,y)

- m

ds

s-k+io r + OO J

F-(k,x,y)

s

MO,*) ^

(3.11)

we can write (3.11) in the form

F-(k, y) = X0(k, y) + (OtF.)(k,

y),

(3.12)

where * ( » . , ) - ¿ A * " 2m J_00 s - k + i0 (O.F-Xk,

y) = -

1

J» + OO J jTj—^p(s)

(3.13)

s e —

y).

Note that, since there axe no bound states, //(0, x) > 0 for all x € R . The next theorem establishes the connection between F-(k,y)

and y(x). Let an overdot denote the

derivative with respect to k. T h e o r e m 3.2 fi(0,x)

Suppose that assumptions (H1)-(H5) hold with a = 1 in (H5). Then

is determined by Q(x) alone and /;(0, x) is determined by Q(x) and H+ alone.

Furthermore, we have - i F - ( 0 , y ) = i ^ \ + y + A+. //(0,x)

(3.14)

PROOF: From (2.3) it follows that /j(0, x) and fi(0, x) obey the integral equations fi(0,x)

= l + J

/i(0,x) = iH+x + j^

dz(z-x)Q(z)f,{0,z), dz(z-x)Q(z)f,(0,z).

(3.15) (3.16)

Eqs. (3.15) and (3.16) can be solved by iteration and the first assertion follows. Eq. (3.14) follows on taking k Note that fi(0,x) Hence F-(Q,y)

0 in (3.8) and using (3.1). |

is purely imaginary and hence the right-hand side of (3.14) is real.

must be purely imaginary. This can also be seen from (3.12) using the

fact that p(k) = p(—k) for k G R . In order to find F-(k,y)

we need to know X0(k, y)

first. We see from (3.13) that Xo(k, y) is completely determined by p(k). Provided (3.12) has a unique solution, F-(k,y)

is also completely determined by p(k). However, there

is the possibility that a restriction on H+ arises from the solution of (3.14), since y(x)

14

Aktosun, Klaus, van der Mee

must also be such that y(x) —• ±00 as x —• ±00. As we will see this situation occurs in Example 2, Section 4. Once F-(k,y)

has been obtained, the value of A.y is determined

by setting x = 0 and y = 0 in (3.14), so that

Then y{x) is found by solving (3.14) for y in terms of x. Finally, H(x) can be obtained by using H(x) =

dy/dx.

We remark that if, in addition to p(k) and Q(x), H- is known instead of H+, then we can first compute H+ as follows. In the exceptional case (i.e. if —1 < p(0) < 1), we get from (2.13) H + =

H

i-P(o)

-

In the generic case (i.e. if p(0) = —1), using (2.8) we first compute

= •y/l — \p(k)\2

for k € R , and then find |c| = limt_>o |r(fc)|/|fc|, where c is the constant given in (2.11). Thus, by (2.11), =j£!Ll[/l(0,«)i/r(0,x)]|

H

2

4 HTheorem 3.2 no longer applies if we only have Q € I 0), and in the exceptional case when Q £ L\(R),

is available [9],

Now we consider the case when there are bound states at energies — k2J with j — 1, • • • , Af. Then the reduced transmission coefficient r(fc) has simple poles on the positive imaginary axis at k = in j. Let dxMiKj,x)2H(x)2>j

,

j = 1,• • • ,J\f,

(3.17)

denote the norming constants. The norming constants are part of the scattering data. We only list here the main steps of the inversion procedure, so that we can apply it in the next section. A detailed derivation is given in Section 8 of [8]. Let

««(*) - ( - i ) ^ n ^ . j=1

=

lK]

w

=

p(k)™(kr\

C 3 - 1 8 )

Inverse Scattering with Discontinuous Wavespeed X(k)

where

= G-(k,

is the analog of

G-(k,x,y)

X

x, y) =

y) - Z,(0,

[Z,(-k,

ky/H(x)

15

y)],

(3.19)

in (3.8). In analogy to (3.12) we have the

F-(k,x,y)

singular integral equation X(k)

= B(k)

fc 1 — (1 /p). Indeed, this is immediate since 2ik

p(k)e

y

k

- p(o) _ f k

\ 0(1/*),

o. *-»±oo.

So it is natural to study (3.12) in H ^ ( R ) . T h e o r e m 3 . 3 For 1 < p < +oo, (3.12) has a unique solution X 6 H ' ( R ) for every Xo € H{1(R). This solution is given by X(k)

=

where the series

converges absolutely in the norm of H{L(R). The proof of this theorem is given in [8] (Theorem 7.1). We add a few remarks about the proof. When p = 2 the result follows from a contraction argument. That Oy is a strict contraction is obvious in the exceptional case, since ||C y || < s u p t e R

< 1.

Moreover, by using p(k) = p(—k) we see that Oy is self-adjoint. As shown in [8], Oy is also a strict contraction in the generic case (when p(0) = —1). To deal with p ^ 2 we derive a (vector) Riemann-Hilbert problem satisfied by any solution of (3.12) which is in H ' ( R ) . The accompanying Riemann-Hilbert problem, where only the asymptotic part

a(k) of p(k) is retained, can be shown to be uniquely solvable by factorization

of an almost periodic 2 x 2 matrix function. It is here where Theorem 2.4, in particular (iv), enters. As a result, (3.12) is a Fredholm integral equation of index zero in H{L(R). A Fredholm argument then leads to the unique solvability of (3.12) in H?L(R), where 1 < p < +oo. As a further result it follows that the spectral radius of Oy is strictly less than one in any space H L ( R ) (1 < p < oo). 4. E x a m p l e s In this section we consider three examples. Since the first two examples have been worked out in detail in [8], we will only state the main results here. The third example is new and we give most of the details. We also comment on the spectrum of Oy in

Inverse Scattering with Discontinuous Wavespeed

17

the first two examples. The spectral properties of Oy in another example can be found in [9]. Here we confine ourselves to constructing H(x) from a given reduced reflection coefficient p(k), H+, and bound state data. The problem where one starts from R{k) requires some additional steps, which are outlined in [8]. In all three examples it is assumed that Q € L ^ R ) , so that Theorem 3.2 applies, and we are allowed to consider the singular integral equations (3.12) and (3.20) in the space H?_(R). Example 1

Suppose that p(k) = p0e^k,

Since p(0) = po

po,P € R ,

|po| < 1.

—1, we are in the exceptional case. We also assume that there are

no bound states. It turns out that the spectrum of Oy consists of the three points —po, 0, and po, each of which is an eigenvalue of infinite multiplicity. So, O y is bounded and self-adjoint, but not compact. The function H(x) is given by

H(x)

=

H+ /i(0,*)2' 1 - Po H+ fi(0,x)2' 1 + P0

X >

Xi,

X
0.

Since p(0) = —1, we are in the generic case. In this case the spectrum of O y consists of the eigenvalue zero and two infinite sequences of eigenvalues that converge to + ( and —respectively. For y < —/3/2, by solving (3.14) we find VW

~

where 0, J A; dk ettk g(k), we obtain

Using the Fourier transform ( T g ) { t ) =

(JF£(-, x, y))(i, x, y) = 2 « p 0 ( x , y)e- K t ,

2y + 0 > 0,

and when /? + 2y < 0, (TB{-,x,y)){t,x,y)

=

/ 2«p0(x,y)e~Kt

+DU

\ 2™ Po (x, y)e~Kt + D2,

0 - ( 2 y + 0),

where n

A»,-« „ - « « ¿ ( M )

ow«

V7i+

V-"+ D2 = 4 « p o e - K , ( l Let

=

M * )

.

= (:FX(-))(i). For the operator Cy in (3.18) we have

that Cy - 0, when 2y + f) > 0, and when 2y + /? < 0 - 2p0Ke-Ki {TCyT^hXt)

={

a

/ due~KUh(u) J-2V-0-t + Po h(-t - 2y - /?), 0 < t < —(2y + ¡3), r-iy-P — 2poKe~Kt e~K(2y+P) I due-™h{u), t > ~{2y + ¡3). Jo c-«
0,

(4.2)

Inverse Scattering with Discontinuous Wavespeed

19

and when 2y + ft < 0

h(t) =2TTip0(x,y)e-Kt+4Trip0e-Kt(l

r-2y-P / du e~KU h(u) Jo

-2pOKe- e~ Kt

-

K(2y+0)

VH+

(4.3)

if t > - ( 2 y + ft), and A(t) =27ripo(®,y)e " +4ttip 0 e

-2nip0

K

VH+ VH+ r-2y-P / du e~KU h{u) + p0h(-t -2y - ft) J-2y-B-t

- 2p0Ke-Kte~K^+^

(4.4)

if 0 < t < —(2y + /?). We first solve (4.4) and then use the result in (4.3). Following [8] (Example 6.2) we can solve (4.4) by converting it to a second-order differential equation for the function JQ' du e~KU h(u). The solution is

h(t)=-2m/°

*0,

— and (1 -

i/i =

U2

po)po

_

uH

1 - poe^y+n

+ c -2k(„+/U) 2k

1 - pae 0,

-2po 2d 1 — po 1 (1 + po)k ' k 1 + Po z[a{\ - pi) + dpo] - d

Mo,') /,(0,x)

i\ 2

1 ~ Po In; 2(1 + p0)k w


1 and (4.11) holds when 0 < z < 1. Differentiating (4.10) and (4.11), using dz/dx = 2kH{x)z H+

H(x)

=

(

az + poa — d\

az — poa + d J

1-/90

H+

and solving for H(x) we get

2

2y + 0 > 0,

'

f z[a( A K i 1- -i pp + dpo]

(4.12) - d

2y + ¡3 < 0.

Note that /j(0, x) has one zero. It can be seen from (4.10) and (4.11) that this zero is canceled by the zero of the numerator in (4.12). The j u m p in H(x) occurs at x i , where y = —/3/2, i.e. z — 1. Then, by (4.12) we have - 0) _ 1 pp H(Xl+ 0) ~ l + po'

H(x

1

When po — 0 we see that H(x) is continuous. This special case is worked out in [8] (Example 8.1).

Inverse Scattering with Discontinuous Wavespeed

21

Acknowledgments. M. K. thanks G. Scharf for a useful discussion. This material is based upon work supported by the National Science Foundation under grant No. DMS9217627, performed under the auspices of C.N.R.-G.N.F.M., and partially supported by the research project "Nonlinear problems in analysis and its physical, chemical and biological applications: analytical, modeling and computational aspects" of the Italian Ministry of Higher Education and Research (M.U.R.S.T.).

References 1. L. D. Faddeev, Properties of the 5-matrix of the one-dimensional Schrodinger equation, Amer. Math. Soc. Transl. 2, 139 (1964) [Trudy Mat. Inst. Steklova 73, 314 (1964) (Russian)]. 2. V. A. Marchenko, Sturm-Liouville operators and applications, Birkhauser, OT 22, Basel (1986). 3. P. Deift and E. Trubowitz, Inverse scattering on the line. Comm. Pure Appl. Math. 32, 121 (1979). 4. T. Aktosun, M. Klaus, and C. van der Mee, Scattering and inverse scattering in one-dimensional nonhomogeneous media, J. Math. Phys. 34i 1717 (1992). 5. T. Aktosun, M. Klaus, and C. van der Mee, Inverse scattering in 1-D nonhomogeneous media and the recovery of the wavespeed, J. Math. Phys. 33, 1395 (1992). 6. N. I. Grinberg, The one-dimensional inverse scattering problem for the wave equation. Math. USSR Sbornik 70, 557 (1991) [ Mat. Sb. 181 (8), 1114 (1990) (Russian)]. 7. N. I. Grinberg, Inverse scattering problem for an elastic layered medium. Inverse Problems 7, 567 (1991). 8. T. Aktosun, M. Klaus, and C. van der Mee, Inverse wave scattering with discontinuous wavespeed. preprint (1994). 9. T. Aktosun, M. Klaus, and C. van der Mee, On the recovery of a discontinuous wavespeed in wave scattering in a nonhomogeneous medium. In: D. Ivanchev and D. Mishev (Eds.), Proceedings of the 20th Summer School "Applications of Mathematics in Engineering," Varna, August 26-September 2 (1994).

5th Int. Coll. on Differential Equations, p. 23-38 D. Bainov and V. Covachev (Eds) © VSP 1995

Difference m e t h o d s for impulsive partial differential equations D . BAINOV1, 1 2

Z. KAMONT2,

E . MINCHEV1, K .

PACZKOWSKA-PRZADKA2

Higher Medical Institute, Sofia 1504, P.O. Box 45, Bulgaria University of Gdansk, Gdansk, Poland

Abstract. The paper deals with difference methods for the Cauchy problem for nonlinear impulsive partial differential equations of first order. We give sufficient conditions for convergence of the sequence {i>h}, where t)/, is a solution of the difference method, to a solution of the Cauchy problem. The problems of the stability of finite difference equations generated by first order partial differential equations are investigated by the difference inequalities methods. Keywords and phrases: difference methods, impulsive partial differential equations

1

Introduction

The theory of impulsive ordinary differential equations marks its beginning with the paper of V. Mil'man and A. Myshkis [1]. This theory was an object of many investigations in the last three decades and now it is considerably developed [2], [3]. In the last years the theory of impulsive partial differential equations started [4], [5], [6], [7]. These papers are devoted to the qualitative theory of the impulsive partial differential equations. It is necessary to investigate numerical methods for this new theory in order to use these for mathematical simulation. Difference methods for the Cauchy problem for partial differential equations of first order were considered by many authors and under various assumptions: [8], [9], [10], [11], [12], [13]. Difference methods for parabolic equations were considered in [14], [15]. The main problem in these investigations is to find such a difference equation which is stable and satisfies a consistency condition with respect to a differential problem. The problems of stability of finite difference equations generated by first order partial differential equations are investigated by difference inequalities methods or by means of comparison theorems for linear recurrent inequalities. In the present paper we investigate the stability of a one-step difference method for the impulsive partial differential equations using a comparison theorem on impulsive difference inequalities. We introduce some general difference operators which enable us to get a general class of difference schemes which are convergent to a solution of the problem under consideration.

24

D. Bainov

2

et al.

Preliminary notes

Denote by C(X, Y) the set of all continuous functions defined on X taking values in Y, where X and Y are arbitrary metric spaces. Let E = {(*,

y) = (x,

G R1+n

yn)

: x G [0, o], y G [-b

+ Mx,

b - Mi]) ,

where b = ( 6 i , . . . , bn),

M = ( M i , . . . , Mn),

a > 0, 6, > 0, Mi > 0, aMi < bit i = 1 , . . . , n.

Suppose that 0 < ai < 02 < . .. < We define

< a are given numbers and a0 = 0, ak+i = a-

r » = { ( s , y) G E : o 4 < x < ai+i}

, i = 0 , 1 , . . . , k,

Ei = { ( x , y) G E : x = a j , i = 0, 1 , . . . , k + 1, = Ex U E2 U . . . U

Eimp

Let Cimp(E,

(1)

Ek.

R) be the class of all functions z : E —» R such that:

(i) the functions 2Ip^ i = 0 , 1 , . . . , k, Z\EQ are continuous, (ii) for each (x, y) € Eit i = l , . . . , f c + l , there exists lim z(t,s) (1 ,s)-(.x,y) tx

(iii) for each (x, y) G Ei, i = 0 , 1 , . . . , k, we have z(x+, y) = z(x, y) and z(a~, y) = z(a, y), y G [—6 + Ma, b Ma], Let f l =

(E

— Eimp)

x

R x Rn,

^limp

=

Eimp

x R.

Suppose that / : ii — >

R,

9 • ^imp * R> tp : [—6, 6] —• R are given functions. For the function z : E —• R of the variables ( x , y) = (x, yi, • • •, yn) we denote: Dyz = (Dyiz,..., DVnz). We consider the Cauchy problem: Dxz{x,y) z(x,

= f(x,y, y) -

z(x~,y)

z{x,y),

Dyz(x,y)),

= g(x,y,

*{o,v)

= v(v),

z(x~,y)),

(x, y) G E - Eimp, (x,y)

G Eimp,

(2) (3) (4)

ye[-b,b}.

Definition 1. A function z : E — • R is said to be a solution of the problem (2)-(4) if: (i) z G Cimp(E,

R),

t h e r e e x i s t t h e d e r i v a t i v e s Dxz(x,

and z satisfies (2) on E — Eimp, (ii) z satisfies (3) and (4).

y), Dyz(x,

y) for ( x , y) G E —

Eimp

Difference methods...

25

We introduce a mesh in E. Let d = (do, d 1, • .., dn) G R1+n, di > 0 for i = 0 , 1 , . . ., n. Assume that for some h = (ho,hi,...,hn) G (0,d] there exist positive integers No, (Ni,..., Nn) = N such that Ngho = a, Nihi = for i = 1 , . . . , n. Denote by Id the subset of all h E (0, d\ having the above property. Let /Vq1', i = 1, . .. , k, be positive integers chosen so that N^h0 < Oj < (N^ + l)/io for i = 1,. .., k. Let x ( m o ) = m0h0 m

y( >) __ m j / l i

for

for m 0 = 0 , 1 , . . ., No-

m j =

0 ) ± 1 , . . . I ±Ni,

i = 1 , . . . , n. and j/ ( m , ) = ( j / i m i ) , . . . , y

For m = (m 0 , m i , . . . , m n ) we denote m' = ( m i , . . . , mn) Let \h\ = h0 + hx + ... + hn. We define the following sets for h G Id' Eh={(x, y ">). We define the difference operators A, B , So, S = (5i,. . . , 6„) in the following way: AzW

= a0z(m)

+ £ ¿=i

Bz{m)

= po z(m)

+ ¿/?i[*(i(m)) + ¿=1

+ *(-«"*»],

(5)

(6) where a^, Pi € R, i = 0, 1,. . . , n,

Soz {m} = /io 1 [* ( m o + 1 ' m ' ) - Az ( m ) \

(7)

Si2(m> = (2/i i )" 1 [^ ( i ( m ) ) - -z ( ~ i(m)) ], » = 1 , . . . , n.

(8)

mp

Suppose that fh : —> R, g^ : f25, —> R, R and hi < Mih0,

T h e o r e m 1. Suppose 1° Assumption

E?*,

Hi is

i = 1,. . . , n.

that: satisfied.

2° u, v : Eh —> R, u, v satisfy the difference 5 0 u ( m ) - fh(x{mo),

inequalities

y(m'\ B u ( m ) , 5 « ( m ) ) < (13)

< 50v(m) the inequalities

- fh(x(mo\

for the u (m 0 +l,m')

y(m'\ Bv^m\ 5v)

impulses _ u(m) _ gh(x(™o)f j K l ^ W )

< „ ( m 0 + i y ) _ v(m) _ gh(x(mo)! and the initial

y(m')^

E™p,


— There are two cases % p to be distinguished. On Eh — E ™ we derive following estimation: w(mo+l,m')

+ho [fh(x(mo\

_ u(mo+l,m') _ ^(mo+l.ra') < Au^

y{m'\ Bu{m\

—

5u ( m ) ) - / h ( x ( m o ) , y{m'\ Bv{m\

5v ( m ) )] =

n

= Aw(m> + h0Dpfh(P)BwW = wWh

+

+ h0J2DqJh(P)Siw(m^ ¿=1

+

=

h0p0Dpfh(P)]+

n + J2w(i(m)) ¿=1 +

J2w(-i(m)) i=1

h

+ hoPiDpfh(P)

JQ. +

+ h0(2hi)-1DqJh(P)} - ho(2hi)~1DqJh(P)]

hopiDpjh(P)

+• < 0,

where P is an intermediate point. On El™v we obtain w(mo+l,m')

_ u(m0+l,m') _ v(m0+l,m')


o,

Oi + hoPiDpfh(P)

+ h0hTlDqJh{P)

> o,

for i G /(+>,

on + hoPiDpfh{P)

- h0hrxDqJh{P)

> 0, for i e

28

D. Bainov et al.

T h e o r e m 2. Suppose

that:

1° Assumption

satisfied.

Hi

is

2° u, v : Eh —> R, u, v satisfy the difference inequalities (13), where the difference operators S are replaced by the difference operators 8 defined by (17), u, v satisfy the inequalities for the impulses u(mo+l,m')

< v(m°+1'm') and the initial

_ u(m) _ ^ ( m „ ) _ y(m')_ u (m))
(m) on The proof is similar to the proof of Theorem 1. We omit the details.

3.2

Stability of t h e impulsive difference m e t h o d

We introduce Assumption H2. Suppose that: 1° there exists a constant L such that \Dpfh(x,y,p,q)\ \9h{x,y,v)

onflft,

< L

< L0\p-p\

~ 9h(x,y,p)|

on

2° a 0 + 2 E on = 1, A, + 2 E pi = 1, i= 1 t=l 3°

- ) | < a„,k for -N R, Vh is a solution of the impulsive |v(0,m') _ ~(0,m')| < aoA; -N