Page 268 - Handbook Of Integral Equations

Chapter 4

Linear Equations of the Second Kind

With Constant Limits of Integration

Notation: f = f(x), g = g(x), h = h(x), v = v(x), w = w(x), K = K(x) are arbitrary functions;
A, B, C, D, E, a, b, c, l, α, β, γ, δ, µ, and ν are arbitrary parameters; n is a nonnegative integer;
and i is the imaginary unit.

Preliminary remarks. A number λ is called a characteristic value of the integral equation

b
y(x)– λ K(x, t)y(t) dt = f(x)
a
if there exist nontrivial solutions of the corresponding homogeneous equation (with f(x) ≡ 0). The
nontrivial solutions themselves are called the eigenfunctions of the integral equation corresponding to
the characteristic value λ.If λ is a characteristic value, the number 1/λ is called an eigenvalue of the
integral equation. A value of the parameter λ is said to be regular if for this value the homogeneous
equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions
of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the
kernel K(x, t). In the above equation, it is usually assumed that a ≤ x ≤ b.

4.1. Equations Whose Kernels Contain Power-Law
Functions

4.1-1. Kernels Linear in the Arguments x and t

b
1. y(x)– λ (x – t)y(t) dt = f(x).
a
Solution:
y(x)= f(x)+ λ(A 1 x + A 2 ),
where

12f 1 +6λ (f 1 ∆ 2 –2f 2 ∆ 1 ) –12f 2 +2λ (3f 2 ∆ 2 –2f 1 ∆ 3 )
A 1 = , A 2 = ,
4
2
4
2
λ ∆ +12 λ ∆ +12
1 1

b b
n
n
f 1 = f(x) dx, f 2 = xf(x) dx, ∆ n = b – a .
a a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 247