Page 607 - Handbook Of Integral Equations

where the x j are the nodes of the quadrature formula, the A j are given coefficients that do not
depend on the function ϕ(x), and ε n [ϕ] is the error of replacement of the integral by the sum (the
truncation error).
If in the Fredholm integral equation of the second kind,
b
y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (2)
a
we set x = x i (i =1, ... , n), then we obtain the following relation that is the basic formula for the
method under consideration:
b

y(x i ) – λ K(x i , t)y(t) dt = f(x i ), i =1, ... , n. (3)
a
Applying the quadrature formula (1) to the integral in (3), we arrive at the following system of
equations:
n

y(x i ) – λ A j K(x i , x j )y(x j )= f(x i )+ λε n [y]. (4)
j=1
By neglecting the small term λε n [y] in this formula, we obtain the system of linear algebraic
equations for approximate values y i of the solution y(x) at the nodes x 1 , ... , x n :
n

y i – λ A j K ij y j = f i , i =1, ... , n, (5)
j=1
where K ij = K(x i , x j ), f i = f(x i ).
The solution of system (5) gives the values y 1 , ... , y n , which determine an approximate solution
of the integral equation (2) on the entire interval [a, b] by interpolation. Here for the approximate
solution we can take the function obtained by linear interpolation, i.e., the function that coincides
with y i at the points x i and is linear on each of the intervals [x i , x i+1 ]. Moreover, for an analytic
expression of the approximate solution to the equation, a function
n

˜ y(x)= f(x)+ λ A j K(x, x j )y j (6)
j=1
can be chosen, which also takes the values y 1 , ... , y n at the points x 1 , ... , x n .

11.18-2. Construction of the Eigenfunctions
The method of quadratures can also be applied for solutions of homogeneous Fredholm equations
of the second kind. In this case, system (5) becomes homogeneous (f i = 0) and has a nontrivial
solution only if its determinant D(λ) is equal to zero. The algebraic equation D(λ) = 0 of degree n
˜
˜
for λ makes it possible to find the roots λ 1 , ... , λ n , which are approximate values of n characteristic
˜
values of the equation. The substitution of each value λ k (k =1, ... , n) into (5) for f i ≡ 0 leads to
the system of equations
n

(k) ˜ (k)
y i – λ k A j K ij y j =0, i =1, ... , n,
j=1
(k)
whose nonzero solutions y i make it possible to obtain approximate expressions for the eigenfunc-
tions of the integral equation:
n

˜ (k)
˜ y k (x)= λ k A j K(x, x j )y .
j
j=1
˜
If λ differs from each of the roots λ k , then the nonhomogeneous system of linear algebraic
equations (5) has a unique solution. In the same case, the homogeneous system of equations (5) has
only the trivial solution.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 590