Page 606 - Handbook Of Integral Equations

Example. Let us find the first two characteristic values of the integral equation
1

ε[y(x)] ≡ y(x) – λ K(x, t)y(t) dt =0,
0
where
t for t ≤ x,
"
K(x, t)= (5)
x for t > x.
On the basis of (5), we have
x 1

ε[y(x)] = y(x) – λ ty(t) dt + xy(t) dt .
0 x
2
We set Y 2 (x)= A 1 x + A 2 x . In this case
3

4
2
3
ε[Y 2 (x)] = A 1 x + A 2 x – λ 1 A 1 x + 1 A 2 x + x 1 A 1 + 1 A 2 – 1 A 1 x + 1 A 2 x 4 =
3 4 2 3 2 3
1 1 3 1 2 1 4
= A 1 1 – λ x + λx + A 2 – λx + x + λx .
2 6 3 12
On orthogonalizing the residual ε[Y 2 (x)], we obtain the system
1

ε[Y 2 (x)]xdx =0,
0
1

2
ε[Y 2 (x)]x dx =0,
0
or the following homogeneous system of two algebraic equations with two unknowns:
A 1 (120 – 48λ)+ A 2 (90 – 35λ)=0
(6)
A 1 (630 – 245λ)+ A 2 (504 – 180λ)=0.
On equating the determinant of system (6) with zero, we obtain the following equation for the characteristic values:
120 – 48λ 90 – 35λ

630 – 245λ 504 – 180λ
D(λ) ≡ =0.
Hence,
2
λ – 26.03λ + 58.15 = 0. (7)
Equations (7) imply
˜ λ 1 = 2.462 ... and ˜ λ 2 = 23.568 ...
For comparison we present the exact characteristic values:
2
2
λ 1 = 1 π = 2.467 ... and λ 2 = 9 π = 22.206 ... ,
4 4
which can be obtained from the solution of the following boundary value problem equivalent to the original equation:
y xx (x)+ λy(x)=0; y(0)=0, y x (1)=0.

Thus, the error of ˜ λ 1 is approximately equal to 0.2% and that of ˜ λ 2 ,to6%.
•
References for Section 11.17: L. V. Kantorovich and V. I. Krylov (1958), B. P. Demidovich, I. A. Maron, and
E. Z. Shuvalova (1963), A. F. Verlan’ and V. S. Sizikov (1986).
11.18. The Quadrature Method
11.18-1. The General Scheme for Fredholm Equations of the Second Kind
In the solution of an integral equation, the reduction to the solution of systems of algebraic equations
obtained by replacing the integrals with finite sums is one of the most effective tools. The method
of quadratures is related to the approximation methods. It is widespread in practice because it is
rather universal with respect to the principle of constructing algorithms for solving both linear and
nonlinear equations.
Just as in the case of Volterra equations, the method is based on a quadrature formula (see
Subsection 8.7-1):
b n

ϕ(x) dx = A j ϕ(x j )+ ε n [ϕ], (1)
a j=1

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 589