Page 601 - Handbook Of Integral Equations

11.15-2. The Approximate Solution

We set
n

Y n (x)= ϕ 0 (x)+ A i ϕ i (x), (5)
i=1
where ϕ 0 (x), ϕ 1 (x), ... , ϕ n (x) are given functions (coordinate functions) and A 1 , ... , A n are
indeterminate coefficients, and assume that the functions ϕ i (x)(i=1, ... , n) are linearly independent.
Note that, in particular, we can take ϕ 0 (x)= f(x)or ϕ 0 (x) ≡ 0. On substituting the expression (5)
into the left-hand side of Eq. (1), we obtain the residual

n b n

ε[Y n (x)] = ϕ 0 (x)+ A i ϕ i (x) – f(x) – λ K(x, t) ϕ 0 (t)+ A i ϕ i (t) dt,
a
i=1 i=1
or
n

ε[Y n (x)] = ψ 0 (x, λ)+ A i ψ i (x, λ), (6)
i=1
where
b
ψ 0 (x, λ)= ϕ 0 (x) – f(x) – λ K(x, t)ϕ 0 (t) dt
a
b (7)
ψ i (x, λ)= ϕ i (x) – λ K(x, t)ϕ i (t) dt, i =1, ... , n.
a
According to the collocation method, we require that the residual ε[Y n (x)] be zero at the given
system of the collocation points x 1 , ... , x n on the interval [a, b], i.e., we set

ε[Y n (x j )] = 0, j =1, ... , n,

where
a ≤ x 1 < x 2 < ··· < x n–1 < x n ≤ b.

It is common practice to set x 1 = a and x n = b.
This, together with formula (6) implies the linear algebraic system

n

A i ψ i (x j , λ)= –ψ 0 (x j , λ), j =1, ... , n, (8)
i=1
for the coefficients A 1 , ... , A n . If the determinant of system (8) is nonzero,

ψ 1 (x 1 , λ) ψ 1 (x 2 , λ) ··· ψ 1 (x n , λ)

ψ 2 (x 1 , λ) ψ 2 (x 2 , λ) ··· ψ 2 (x n , λ)

det[ψ i (x j , λ)] = . . . . ≠ 0,
. . . . .
. . .

ψ n (x 1 , λ) ψ n (x 2 , λ) ··· ψ n (x n , λ)
then system (8) uniquely determines the numbers A 1 , ... , A n , and hence makes it possible to find
the approximate solution Y n (x) by formula (5).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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