Page 474 - Handbook Of Integral Equations

where K ij = K(x i , x j )(j =1, ... , i), f i = f(x i ), and y j are approximate values of the unknown
function at the nodes x i .
Now system (10) permits one, provided that A ii K ii ≠ 0(i =2, ... , n), to successively find the
desired approximate values by the formulas
n–1

f n – A nj K nj y j
f (a) f 2 – A 21 K 21 y 1 j=1

x
y 1 = , y 2 = , ... , y n = ,
K 11 A 22 K 22 A nn K nn
whose specific form depends on the choice of the quadrature formula.

8.7-3. An Algorithm Based on the Trapezoidal Rule
According to the trapezoidal rule (3), we have

1
A i1 = A ii = h, A i2 = ··· = A i,i–1 = h, i =2, ... , n.
2
The application of the trapezoidal rule in the general scheme leads to the following step algorithm:

f (a) –3f 1 +4f 2 – f 3
x

y 1 = , f (a)= ,
x
K 11 2h
i–1 1

2 f i for j =1,
y i = – β j K ij y j , β j = 2 i =2, ... , n,
K ii h 1 for j >1,
j=1
where the notation coincides with that introduced in Subsection 8.7-2. The trapezoidal rule is quite
simple and effective and frequently used in practice for solving integral equations with variable limit
of integration.
On the basis of Subsections 8.7-1 and 8.7-2, one can write out similar expressions for other
quadrature formulas. However, they must be used with care. For example, the application of
Simpson’s rule must be alternated, for odd nodes, with some other rule, e.g., the rectangle rule or
the trapezoidal rule. For equations with variable integration limit, the use of Chebyshev’s formula
or Gauss’ formula also has some difficulties as well.

8.7-4. An Algorithm for an Equation With Degenerate Kernel
A general property of the algorithms of the method of quadratures in the solution of the Volterra
equations of the first kind with arbitrary kernel is that the amount of computational work at each
step is proportional to the number of the step: all operations of the previous step are repeated with
new data and another term in the sum is added.
However, if the kernel in Eq. (7) is degenerate, i.e.,
m

K(x, t)= p k (x)q k (t), (11)
k=1
or if the kernel under consideration can be approximated by a degenerate kernel, then an algorithm can
be constructed for which the number of operations does not depend on the index of the digitalization
node. With regard to (11), Eq. (7) becomes
m x

p k (x) q k (t)y(t) dt = f(x). (12)
a
k=1

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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