Page 218 - Handbook Of Integral Equations

Chapter 3

Linear Equation of the First Kind

With Constant Limits 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, a, b, c,
k, α, β, γ, λ, and µ are free parameters; and n is a nonnegative integer.

3.1. Equations Whose Kernels Contain Power-Law
Functions

3.1-1. Kernels Linear in the Arguments x and t

1
1. |x – t| y(t) dt = f(x).
0
1 . Let us remove the modulus in the integrand:
◦
x 1

(x – t)y(t) dt + (t – x)y(t) dt = f(x). (1)
0 x
Differentiating (1) with respect to x yields

x 1
y(t) dt – y(t) dt = f (x). (2)

x
0 x
Differentiating (2) yields the solution
1

y(x)= f (x). (3)
2 xx
2 . Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy
◦
1

certain relations. By setting x = 0 and x = 1 in (1), we obtain two corollaries ty(t) dt = f(0)
0
1
and (1 – t)y(t) dt = f(1), which can be rewritten in the form
0
1 1
ty(t) dt = f(0), y(t) dt = f(0) + f(1). (4)
0 0
In Section 3.1, we mean that kernels of the integral equations discussed may contain power-law functions or modulus of
power-law functions.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 197