Page 541 - Handbook Of Integral Equations

The first condition in (9) makes it possible to seek the solution of the integral equation (1) as the
sum of the solutions of the integral equations (8), that is,

y(x)= y 1 (x)+ y 2 (x), –1 ≤ x ≤ 1. (10)

Note that by virtue of the last two conditions in (9), the relations
–β 1 x
y 1 (x)= O e as x →∞,
(11)
β 2 x
y 2 (x)= O e as x → –∞,
where β 1 > 0 and β 2 > 0, are valid.
Recall that the kernel K(x) is an even function. Therefore, if f(x) in Eq. (1) is an even or odd
function, then one must set

f 1 (x)= ±f 2 (–x), y 1 (x)= ±y 2 (–x) (12)

in system (8).*
In both cases, system (8) can be reduced by changes of variables to the same integral equation

∞ ∞
K(z – τ)w(τ) dτ = F(z) ± K(2/λ – z – τ)w(τ) dτ, 0 ≤ z < ∞, (13)
0 2/λ
in which the following notation is used:
x +1 t +1 1
z = , τ = , w(τ)= y(t), F(z)= f 1 (x). (14)
λ λ λ
In view of the properties of the kernel K(x) (see Subsection 10.7-1) and the first relation in (11),
the asymptotic estimate

∞
I(w) ≡ K(2/λ – z – τ)w(τ) dτ = O e –2β 1 /λ (15)
2/λ
can be obtained, which is uniform with respect to τ.
According to (15), for small λ the iterative scheme

∞

K(z – τ)w n (τ) dτ = F(z) ± I w n–1 , n =1, 2, ... , (16)
0
can be used to solve the integral equation (13) by the method of successive approximations. In
the main approximation, the integral I(w 0 ) can be omitted on the right-hand side. Equations (16)
are Wiener–Hopf integral equations of the first kind, which can be solved in a closed form (see
Subsection 10.5-1).
It follows from formulas (10), (12), and (14) that, as λ → 0, the leading term of the asymptotic
expansion of the solution of the integral equation (1) has the form

1+ x 1 – x
y(x)= w 1 ± w 1 , (17)
λ λ
where w 1 = w 1 (τ) is the solution of Eq. (16) with n = 1 and w 0 ≡ 0.
For practical purposes, formula (17) is usually sufficient.
* In formulas (12), (13), (16), and (17), the plus sign corresponds to even f(x) and the minus sign to odd f(x).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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