Page 472 - Handbook Of Integral Equations

We set a = e and obtain
C
x e 1
∞ ν –Cν

L dν = . (12)
0 Γ(ν +1) p (ln p + C)
Let us proceed with relation (9). By (12), we have
ν –Cν
f (0) ∞ x e

x = L f (0) dν . (13)

p (ln p + C) x 0 Γ(ν +1)
Taking into account (10) and (12), we can regard the first summand on the right-hand side in (9) as
a product of transforms. To find this summand itself we apply the convolution theorem:
2 ˜
ν –Cν
p f(p) – f (0) x ∞ (x – t) e

x = L f (t) dν dt . (14)

p (ln p + C) 0 tt 0 Γ(ν +1)
On the basis of relations (9), (13), and (14) we obtain the solution of the integral equation (4) in
the form
x
ν –Cν
ν –Cν
∞ (x – t) e ∞ x e
y(x)= – f (t) dν dt – f (0) dν. (15)

tt x
0 0 Γ(ν +1) 0 Γ(ν +1)
•
References for Section 8.6: V. Volterra (1959), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971).
8.7. Method of Quadratures
8.7-1. Quadrature Formulas
The method of quadratures is a method for constructing an approximate solution of an integral
equation based on the replacement of integrals by finite sums according to some formula. Such
formulas are called quadrature formulas and, in general, have the form
b n

ψ(x) dx = A i ψ(x i )+ ε n [ψ], (1)
a i=1
where x i (i =1, ... , n) are the abscissas of the partition points of the integration interval [a, b], or
quadrature (interpolation) nodes, A i (i =1, ... , n) are numerical coefficients independent of the
choice of the function ψ(x), and ε n [ψ] is the remainder (the truncation error) of formula (1). As a
rule, A i ≥ 0 and n A i = b – a.
i=1
There are quite a few quadrature formulas of the form (1). The following formulas are the
simplest and most frequently used in practice.
Rectangle rule:
A 1 = A 2 = ··· = A n–1 = h, A n =0,
b – a (2)
h = , x i = a + h(i – 1) (i =1, ... , n).
n – 1
Trapezoidal rule:
1
A 1 = A n = h, A 2 = A 3 = ··· = A n–1 = h,
2
b – a (3)
h = , x i = a + h(i – 1) (i =1, ... , n).
n – 1
Simpson’s rule (or prizmoidal formula):
1
4
2
A 1 = A 2m+1 = h, A 2 = ··· = A 2m = h, A 3 = ··· = A 2m–1 = h,
3 3 3
b – a (4)
h = , x i = a + h(i – 1) (n =2m +1, i =1, ... , n),
n – 1
where m is a positive integer.
In formulas (2)–(4), h is a constant integration step.
The quadrature formulas due to Chebyshev and Gauss with various numbers of interpolation
nodes are also widely applied. Let us illustrate these formulas by an example.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 454