Integral Equations in Elasticity 0828524416, 9780828524414

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B. 3. IlapTOH, II. H. IlepniiH

HHTErPAJIbHLIE yPABHEHHH

teophh

ynpyrocra

II 3 3 ,ATEnbCTB 0 «HAyKA» • MOCKBA

V, Z. Parton and P. I. Perlin

INTEGRAL EQUATIONS IN ELASTICITY

MIR PUBLISHERS MOSCOW

Translated from the Russian

First published 1982 Revised from the 1977 Russian edition

Ha amjiuuCKOM R3bine

1. Thus, the poles of the resolvent must necessarily coincide with the zeros of the Fredholm determinant. Suppose that k0 is a pole of multiplicity r of the resolvent. We then have the expansion*

(l /f O _

D(X)

a_r(z, y)

( x - x 0y

, a-r+1 f a y ) ,

. «-i (*, I/)

1 ( k - k o V -1( x - x 0)

+ 2 a‘i y) (i-28) i=0 Substituting series (1.28) in the functional equation (1.12), mul­ tiplying it successively by the factors (k — A,0)n (n = r, r — 1, ...), and then setting k = A,0, we obtain the relations ft

a~r(x, y) = K

j

K(x, t) ffl_r (t, y)dt,

a b «-r+1 ( X , y) -----y) = Xo j K

( X,

t) a_r+1 (t, y) dt,

* The parameter k in the coefficients an (x, y) is omitted.

(1.30)

(1.29)

28

ELEMENTS OF THE THEORY OF INTEGRAL EQUATIONS

t1

etc. It follows from relation (1.29) that the coefficient a_r (x, y) as a function of x with an arbitrary fixed value of y (regarded as a parameter) is the solution of the homogeneous equation b

(x) = h0 ^ K ( y , x ) y ( y ) d y ,

(1.31)

a

called the companion (or transposed) equation to (1.1) To construct the complete theory, it is necessary to study the question of the solvability of integral equations on the eigenvalues. It is obvious that the resolvent of the companion equation is obtained from the resolvent of the original equation by transposing the vari­ ables. Consequently, the Fredholm determinants of the original and companion equations are identical, and so are the eigenvalues. Let us prove that the number of eigenfunctions corresponding to the same value X0 is finite (implying linearly independent solu­ tions). Let cp* (:r),cp* (x), • . cpm (x) be orthonormal eigenfunctions corresponding to the number X0. Consider the equalities that are satisfied by these functions: .jj t \

^

— — = j K (x, y) cp| (y) dy.

(1.32)

a

It is obvious that the right-hand side is the Fourier coefficient of the function K (x , y) (as a function of the argument y) in the ortho­ normal system of functions cp* (y). From Bessel’s inequality (m is the number of eigenfunctions) it follows that ™ cpf2(j ) f 2 K H x ' v) dy■ i=l a Remembering that the eigenfunctions are normalized, and inte­ grating both sides of the last inequality with respect to x , we obtain

ANALYTIC THEORY OF A RESOLVENT

1]

29

It follows from this estimate that the number of eigenfunctions is finite. Let us prove that the numbers of eigenfunctions of the original and companion equations (of course, for the same eigenvalue) are equal. Suppose that there are m orthonormal eigenfunctions cp* (x) of the original equation and n functions of the companion equation denoted by if)* (x). Assume that m < n, and consider two companion equations b

m

cp (X) = X0 j [ z (X, y) — 2

y (x) =

(*) Cf| (y)] cp (y) dy,

(1.33)

j [ K (y. *) — 2 'P?(y)vf (* )]'p(y)dy-

(*-34)

a

j —1

b

m

j=1

a

Let us prove that Eq. (1.33) has no eigenfunctions. Multiply this equation by any one of the functions \|)* (x) (/ ^ m) and integrate with respect to x. By interchanging the order of integration in the double integral [keeping in mind the orthonormality of the functions o|?* (#)] we arrive at the equality b

j f) ] = (p(^ t),

4-1 ♦ * » . Tl ) + r ( * . =

f

Jii J

+

+ I M L

L

< * - % ] - - > dT-

(2A6>

Let us consider a special case when the density function cp is a function of only one argument tv It can be shown that the integral on the right-hand side of formula (2.16) vanishes. Then

* Assuming cp to be a function of two arguments.

CAUCHY-TYPE INTEGRAL

2]

45

We now turn to the discussion of the Cauchy-type integral for an unclosed contour. Let L be a smooth unclosed contour with its ends at points a and b. We fix the sense of description, say from the point a to the point 6, and consider the integral Q(z) =

T— Z

dx.

(2.17)

Here the function (p ( t ) belongs to the class H (A, X) at all points of the contour L including the ends. Integral (2.17) will also be termed Cauchy’s integral. This integral, in contrast to integral (2.2), is not a piecewise analytic but an analytic function in the whole plane, with the exception of the contour L. By analogy with the Cauchy-type integral for a closed contour, in the case under con­ sideration we also introduce the concept of a singular value and the concepts of the limiting values from the left and right, ® + (t) and ’

(2'28> 0(t), 7 where O 0 (z) is an analytic function bounded in the neighbourhood of the point a. For the singular value, we have the representa­ tion -(t)] = - ^ p - ( t - a ) - y + % (t ).

(2.29)

We now turn to the consideration of the general case. After simple manipulation, we obtain an expression for the function O (z) in the form of a sum of two integrals: 1 r = a(t) cp (*) +-~r- j M x ^ ^ -