Page 600 - Handbook Of Integral Equations

which can be rewritten in the form
2
(µ – 1)(2µ – 1)(5µ – 1)(13µ – 22µ + 1)=0. (16)
On solving (16), we obtain

˜ λ 1 = 10.02, ˜ λ 2 = 43.2, ˜ λ 3 = 108, ˜ λ 4 = 216, ˜ λ 5 = 355.2.
The exact values of the characteristic values of the equation under consideration are known:
2
2
2
λ 1 = π = 9.869 ... , λ 2 =(2π) = 39.478 ... , λ 3 =(3π) = 88.826 ... ,
and hence the calculation error is 2% for the first characteristic value, 9% for the second characteristic value, and 20% for
the third characteristic value.
The result can be improved by choosing another collection of points x i and t i (i =1, ... , 5). However, for this number
of ordinates we cannot have very high precision, because the kernel Q(x, t) itself has a singularity, namely, its derivative is
discontinuous for x = t, and thus the kernels under consideration cannot provide a good approximation of the given kernel.
•
References for Section 11.14: H. Bateman (1922), E. Goursat (1923), L. V. Kantorovich and V. I. Krylov (1958).

11.15. The Collocation Method

11.15-1. General Remarks

Let us rewrite the Fredholm integral equation of the second kind in the form

b

ε[y(x)] ≡ y(x) – λ K(x, t)y(t) dt – f(x) = 0. (1)
a
Let us seek an approximate solution of Eq. (1) in the special form

Y n (x)= Φ(x, A 1 , ... , A n ) (2)

with free parameters A 1 , ... , A n (undetermined coefficients). On substituting the expression (2)
into Eq. (1), we obtain the residual

b

ε[Y n (x)] = Y n (x) – λ K(x, t)Y n (t) dt – f(x). (3)
a
If y(x) is an exact solution, then, clearly, the residual ε[y(x)] is zero. Therefore, one tries to
choose the parameters A 1 , ... , A n so that, in a sense, the residual ε[Y n (x)] is as small as possible.
The residual ε[Y n (x)] can be minimized in several ways. Usually, to simplify the calculations, a
function Y n (x) linearly depending on the parameters A 1 , ... , A n is taken. On finding the parameters
A 1 , ... , A n , we obtain an approximate solution (2). If

lim Y n (x)= y(x), (4)
n→∞

then, by taking a sufficiently large number of parameters A 1 , ... , A n ,we find that the solution y(x)
can be found with an arbitrary prescribed precision.
Now let us go to the description of a concrete method of construction of an approximate
solution Y n (x).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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