Page 204 - Handbook Of Integral Equations

x

38. y(x) – g(x)+ b cos(λx) – b(x – t)[λ sin(λx) + cos(λx)g(x)] y(t) dt = f(x).
a
This is a special case of equation 2.9.16 with h(x)= b cos(λx).
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) 2 2 H(x) x G(s)
R(x, t)=[g(x)+ b cos(λx)] + b cos (λx) – bλ sin(λx) ds,
G(t) G(t) t H(s)
b
x
where G(x)=exp g(s) ds and H(x)=exp sin(λx) .
a λ
x

39. y(x) – g(x)+ b sin(λx)+ b(x – t)[λ cos(λx) – sin(λx)g(x)] y(t) dt = f(x).
a
This is a special case of equation 2.9.16 with h(x)= b sin(λx).
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) 2 2 H(x) x G(s)
R(x, t)=[g(x)+ b sin(λx)] + b sin (λx)+ bλ cos(λx) ds,
G(t) G(t) H(s)
t
x b

where G(x)=exp g(s) ds and H(x)=exp – cos(λx) .
λ
a

2.9-2. Equations With Difference Kernel: K(x, t)= K(x – t)

x

40. y(x)+ K(x – t)y(t) dt = f(x).
a
Renewal equation.
◦
1 . To solve this integral equation, direct and inverse Laplace transforms are used.
The solution can be represented in the form

x
y(x)= f(x) – R(x – t)f(t) dt. (1)
a

Here the resolvent R(x) is expressed via the kernel K(x) of the original equation as follows:

1 c+i∞ px
˜
R(x)= R(p)e dp,
2πi
c–i∞
˜
K(p) ∞ –px
˜
˜
R(p)= , K(p)= K(x)e dx.
˜
1+ K(p) 0
•
References: R. Bellman and K. L. Cooke (1963), M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971),
V. I. Smirnov (1974).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 183