Page 540 - Handbook Of Integral Equations

To construct an asymptotic solution of Eq. (1) as λ →∞, it is convenient to do the following.
First, we consider the auxiliary integral equation

1
K(x – t, β, λ)y(t) dt = f(x), –1 ≤ x ≤ 1,
–1
(5)
∞ ∞

a n n b n n
K(x, β, λ)= ln |x| – β |x| + |x| ,
λ n λ n
n=0 n=0
with two parameters λ and β. We seek its solution in the form of a regular asymptotic expansion in
negative powers of λ (for fixed β). That is, we have

N
–n –N
y(x, β, λ)= λ y n (x, β)+ o λ . (6)
n=0
Substituting (6) into (5) yields a recurrent chain of integral equations of the form (4):

1

a 0 ln |x – t| – a 0 β + b 0 y n (t, β) dt = g n (x, β), –1 ≤ x ≤ 1, (7)
–1
from which the functions y n (x, β) can be successively calculated. The right-hand sides g n (x, β)
depend only on the previously determined functions y 0 , y 1 , ... , y n–1 .
Note that for β =ln λ the auxiliary equation (5) coincides with the original equation (1) into
which the expansion (3) is substituted. Therefore, the asymptotic solution of Eq. (1) can be obtained
with the aid of (6) and (7) with β =ln λ.
Some contact problems of elasticity can be reduced to Eq. (1), in which the kernel can be
represented in the form (3) with a n = 0 for all n > 0 and b 2m+1 = 0 for m =0, 1, 2, ... In this case,
one must set y n (x, β) ≡ 0(n =1, 3, 5, ... ) in the solution (6). In practice, it usually suffices to
–4
retain the terms up to λ .

10.7-3. The Solution for Small λ
In analyzing the limit case λ → 0, we take into account the singularities of the solution at the
endpoints of the interval –1 ≤ x ≤ 1 (see formula (2)). Consider the following auxiliary system of
two integral equations:

–1
∞ x – t x – t
K y 1 (t) dt = f 1 (x)+ K y 2 (t) dt, –1 ≤ x < ∞,
–1 λ –∞ λ
1 x – t ∞ x – t (8)
K y 2 (t) dt = f 2 (x)+ K y 1 (t) dt, –∞ < x ≤ 1.
λ λ
–∞ 1
The former equation provides for selecting the singularity at x = –1 and the latter for selecting the
singularity at x = +1.
The functions f 1 (x) and f 2 (x) are such that

f 1 (x)+ f 2 (x)= f(x), –1 ≤ x ≤ 1,
–α 1 x
f 1 (x)= O e as x →∞, (9)
α 2 x
f 2 (x)= O e as x → –∞,
where α 1 > 0 and α 2 >0.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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