Page 512 - Handbook Of Integral Equations

10.2-2. Solution of the Main Equation

For any continuous function f(x), the solution of the original equation (1) can be expressed via the
solution of the auxiliary equation (2) by the formula

a
1 d

y(x)= w(t, a)f(t) dt w(x, a)
2M (a) da

–a
a ξ
1 d 1 d
– w(x, ξ) w(t, ξ)f(t) dt dξ (3)
2 |x| dξ M (ξ) dξ –ξ

a ξ
1 d w(x, ξ)
– w(t, ξ) df(t) dξ,
2 dx |x| M (ξ) –ξ

ξ
where M(ξ)= w(x, ξ) dx, the prime stands for the derivative, and the last inner integral is treated
0
as a Stieltjes integral.
Formula (3) permits one to obtain some exact solutions of integral equations of the form (1)
with arbitrary right-hand side, see Section 3.8 of the first part of the book.
Example 1. The solution of the integral equation
a A

ln y(t) dt = f(x),
–a |x – t|
which arises in elasticity, is given by formula (3), where
2A M(ξ)

–1
M(ξ)= ln , w(t, ξ)= .
ξ π ξ – t 2
2
Example 2. Consider the integral equation
a y(t) dt

–a |x – t| µ = f(x), 0 < µ <1,
which arises in the theory of elasticity. The solution is given by formula (3), where
√
2 π 1 πµ
µ–1

2
µ
M(ξ)=
ξ , w(t, ξ)= cos ξ – t 2 2 .
µ
1 – µ π 2
µ Γ Γ
2 2
•
References for Section 10.2: N. Kh. Arutyunyan (1959), I. C. Gohberg and M. G. Krein (1967).
10.3. The Method of Integral Transforms
The method of integral transforms enables one to reduce some integral equations on the entire
axis and on the semiaxis to algebraic equations for transforms. These algebraic equations can readily
be solved for the transform of the desired function. The solution of the original integral equation is
then obtained by applying the inverse integral transform.

10.3-1. Equation With Difference Kernel on the Entire Axis
Consider the integral equation

∞
K(x – t)y(t) dt = f(x), –∞ < x < ∞, (1)
–∞
where f(x), y(x) ∈ L 2 (–∞, ∞) and K(x) ∈ L 1 (–∞, ∞).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 494