Page 679 - Handbook Of Integral Equations

Example 1. Consider the integral equation
x

m
y(x – t)y(t) dt = Ax , m > –1.
0
m
Applying the Laplace transform to the equation under consideration with regard to the relation L{x } = Γ(m +1)p –m–1 ,
we obtain
2
˜ y (p)= AΓ(m +1)p –m–1 ,
where Γ(m) is the gamma function. On extracting the square root of both sides of the equation, we obtain
– m+1
˜ y(p)= ± AΓ(m +1)p 2 .
Applying the Laplace inversion formula, we obtain two solutions to the original integral equation
√ √
AΓ(m +1) m–1 AΓ(m +1) m–1
y 1 (x)= – x 2 , y 2 (x)= x 2 .
m +1 m +1
Γ Γ
2 2
14.2-2. The Method of Differentiation for Integral Equations

Sometimes, differentiation (possibly multiple) of a nonlinear integral equation with subsequent
elimination of the integral terms by means of the original equation makes it possible to reduce this
equation to a nonlinear ordinary differential equation. Below we briefly list some equations of this
type.
1 . The equation
◦
x

y(x)+ f t, y(t) dt = g(x) (3)
a
can be reduced by differentiation to the nonlinear first-order equation

y + f(x, y) – g (x)=0
x
x
with the initial condition y(a)= g(a).
2 . The equation
◦
x

y(x)+ (x – t)f t, y(t) dt = g(x) (4)
a
can be reduced by double differentiation (with the subsequent elimination of the integral term by
using the original equation) to the nonlinear second-order equation:
y + f(x, y) – g (x) = 0. (5)

xx
xx
The initial conditions for the function y = y(x) have the form

y(a)= g(a), y (a)= g (a). (6)
x
x
3 . The equation
◦
x

y(x)+ e λ(x–t) f t, y(t) dt = g(x) (7)
a
can be reduced by differentiation to the nonlinear first-order equation

y + f(x, y) – λy + λg(x) – g (x)=0. (8)
x
x
The desired function y = y(x) must satisfy the initial condition y(a)= g(a).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 662