Page 605 - Handbook Of Integral Equations

11.17. The Bubnov–Galerkin Method

11.17-1. Description of the Method
Let
b

ε[y(x)] ≡ y(x) – λ K(x, t)y(t) dt – f(x) = 0. (1)
a
Similarly to the above reasoning, we seek an approximate solution of Eq. (1) in the form of a finite
sum
n

Y n (x)= f(x)+ A i ϕ i (x), i =1, ... , n, (2)
i=1
where the ϕ i (x)(i =1, ... , n) are some given linearly independent functions (coordinate functions)
and A 1 , ... , A n are indeterminate coefficients. On substituting the expression (2) into the left-hand
side of Eq. (1), we obtain the residual
n b b

ε[Y n (x)] = A j ϕ j (x) – λ K(x, t)ϕ j (t) dt – λ K(x, t)f(t) dt. (3)
a a
j=1
According to the Bubnov–Galerkin method, the coefficients A i (i =1, ... , n) are defined from
the condition that the residual is orthogonal to all coordinate functions ϕ 1 (x), ... , ϕ n (x). This gives
the system of equations
b
ε[Y n (x)]ϕ i (x) dx =0, i =1, ... , n,
a
or, by virtue of (3),
n

(α ij – λβ ij )A j = λγ i , i =1, ... , n, (4)
j=1
where
b b b b b

α ij = ϕ i (x)ϕ j (x) dx, β ij = K(x, t)ϕ i (x)ϕ j (t) dt dx, γ i = K(x, t)ϕ i (x)f(t) dt dx.
a a a a a
If the determinant of system (4)
D(λ) = det[α ij – λβ ij ]
is nonzero, then this system uniquely determines the coefficients A 1 , ... , A n . In this case, formula (2)
gives an approximate solution of the integral equation (1).

11.17-2. Characteristic Values
˜
˜
The equation D(λ) = 0 gives approximate characteristic values λ 1 , ... , λ n of the integral equation.
On finding nonzero solutions of the homogeneous linear system
n
(k)
˜
(α ij – λ k β ij )A ˜ j =0, i =1, ... , n,
j=1
˜
we can construct approximate eigenfunctions Y ˜ (k) (x) corresponding to characteristic values λ k :
n
n
(k)
˜
Y ˜ (k) (x)= A ϕ(x).
i
n
i=1
It can be shown that the Bubnov–Galerkin method is equivalent to the replacement of the
kernel K(x, t) by some degenerate kernel K (n) (x, t). Therefore, for the approximate solution Y n (x)
we have an error estimate similar to that presented in Subsection 11.13-2.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 588