Page 152 - Handbook Of Integral Equations

–1
where L R(p) is the inverse Laplace transform of the function
n
K(p) A k
R(p)= , K(p)= . (11)
1+ K(p) p – λ k
k=1
The transform R(p) of the resolvent R(x) can be represented as a regular fractional
function:
Q(p)
R(p)= , P(p)=(p – µ 1 )(p – µ 2 ) ... (p – µ n ),
P(p)
where Q(p) is a polynomial in p of degree < n. The roots µ k of the polynomial P(p) coincide
with the roots of equation (8). If all µ k are real and different, then the resolvent can be
determined by the formula

n
µ k x Q(µ k )
R(x)= B k e , B k = ,
P (µ k )

k=1
where the prime stands for differentiation.

2.2-2. Kernels Containing Power-Law and Exponential Functions

x
20. y(x)+ A xe λ(x–t) y(t) dt = f(x).
a
Solution:
x
2
2
y(x)= f(x) – A x exp 1 A(t – x )+ λ(x – t) f(t) dt.

2
a
x

21. y(x)+ A te λ(x–t) y(t) dt = f(x).
a
Solution:
x

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

λt
22. y(x)+ A (x – t)e y(t) dt = f(x).
a
λt
This is a special case of equation 2.9.4 with g(t)= Ae .
Solution:
A x λt
y(x)= f(x)+ u 1 (x)u 2 (t) – u 2 (x)u 1 (t) e f(t) dt,
W a
where u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear homo-
λx
geneous ordinary differential equation u + Ae u = 0; the functions u 1 (x) and u 2 (x) are
xx
expressed in terms of Bessel functions or modified Bessel functions, depending on sign A:
√ √
λ 2 A λx/2 2 A λx/2
W = , u 1 (x)= J 0 e , u 2 (x)= Y 0 e for A >0,
π λ λ
√ √
λ 2 |A| λx/2 2 |A| λx/2
W = – , u 1 (x)= I 0 e , u 2 (x)= K 0 e for A <0.
2 λ λ
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 131