Page 128 - Handbook Of Integral Equations

Chapter 2

Linear Equations of the Second Kind

With Variable Limit of Integration

Notation: f = f(x), g = g(x), h = h(x), K = K(x), and M = M(x) are arbitrary functions (these
may be composite functions of the argument depending on two variables x and t); A, B, C, D, a,
b, c, α, β, γ, λ, and µ are free parameters; and m and n are nonnegative integers.

2.1. Equations Whose Kernels Contain Power-Law
Functions

2.1-1. Kernels Linear in the Arguments x and t

x
1. y(x)– λ y(t) dt = f(x).
a
Solution:
x
y(x)= f(x)+ λ e λ(x–t) f(t) dt.
a
x
2. y(x)+ λx y(t) dt = f(x).
a
Solution:
x

1 2 2
y(x)= f(x)– λ x exp λ(t – x ) f(t) dt.
2
a
x

3. y(x)+ λ ty(t) dt = f(x).
a
Solution:
x

1 2 2
y(x)= f(x)– λ t exp λ(t – x ) f(t) dt.
2
a
x
4. y(x)+ λ (x – t)y(t) dt = f(x).
a
This is a special case of equation 2.1.34 with n =1.
1 . Solution with λ >0:
◦
x √
y(x)= f(x)– k sin[k(x – t)]f(t) dt, k = λ.
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 107