Page 173 - Handbook Of Integral Equations

Eliminating the integral I n from (4) and (5), we obtain

n–1
2
2
2
2
y (x)+(σ n + λ )y(x)+ A k (λ – λ )I k = f (x)+ λ f(x). (6)

xx n n k xx n
k=1
Differentiating (6) with respect to x twice followed by eliminating I n–1 from the resulting
expression with the aid of (6) yields a similar equation whose left-hand side is a fourth-
n–2
order differential operator (acting on y) with constant coefficients plus the sum B k I k .
k=1
Successively eliminating the terms I n–2 , I n–3 , ... using double differentiation and formula (3),
we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with
constant coefficients.
The initial conditions for y(x) can be obtained by setting x = a in the integral equation
and all its derivative equations.
2 . Let us find the roots z k of the algebraic equation
◦
n
λ k A k

+ 1 = 0. (7)
z + λ 2
k=1 k
By reducing it to a common denominator, we arrive at the problem of determining the roots
of an nth-degree characteristic polynomial.
Assume that all z k are real, different, and nonzero. Let us divide the roots into two groups
z 1 >0, z 2 >0, ... , z s > 0 (positive roots);
z s+1 <0, z s+2 <0, ... , z n < 0 (negative roots).
Then the solution of the integral equation can be written in the form

s n
x

y(x)=f(x)+ B k sinh µ k (x–t) + C k sin µ k (x–t) f(t) dt, µ k = |z k |. (8)
a
k=1 k=s+1
The coefficients B k and C k are determined from the following system of linear algebraic
equations:
s n

B k µ k C k µ k
+ – 1=0, µ k = |z k | m =1, 2, ... , n. (9)
2
2
λ + µ 2 λ – µ 2
m
m
k=0 k k=s+1 k
In the case of a nonzero root z s = 0, we can introduce the new constant D = B s µ s and
proceed to the limit µ s → 0. As a result, the term D(x – t) appears in solution (8) instead of
–2
B s sinh µ s (x – t) and the corresponding terms Dλ appear in system (9).
m
x
sin(λx)
20. y(x) – A y(t) dt = f(x).
a sin(λt)
Solution:
x sin(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a sin(λt)
x
sin(λt)
21. y(x) – A y(t) dt = f(x).
a sin(λx)
Solution:
x sin(λt)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a sin(λx)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 152