Page 463 - Handbook Of Integral Equations

8.3-2. The Second Method
Let us introduce the new variable
x

Y (x)= y(t) dt
a
and integrate the right-hand side of Eq. (1) by parts taking into account the relation f(a) = 0. After
dividing the resulting expression by K(x, x), we arrive at the Volterra equation of the second kind
x K (x, t) f(x)

Y (x) – t Y (t) dt = ,
a K(x, x) K(x, x)
for which the condition K(x, x) /≡ 0 must hold.
•
References for Section 8.3: E. Goursat (1923), V. Volterra (1959).

8.4. Equations With Difference Kernel: K(x, t)= K(x – t)

8.4-1. A Solution Method Based on the Laplace Transform
Volterra equations of the first kind with kernel depending on the difference of the arguments have
the form
x

K(x – t)y(t) dt = f(x). (1)
0
To solve these equations, the Laplace transform can be used (see Section 7.2). In what follows
we need the transforms of the kernel and the right-hand side; they are given by the formulas
∞ ∞

˜
˜
K(p)= K(x)e –px dx, f(p)= f(x)e –px dx. (2)
0 0
Applying the Laplace transform L to Eq. (1) and taking into account the fact that an integral
with kernel depending on the difference of the arguments is transformed to the product by the rule
(see Subsection 7.2-3)
x
˜
L K(x – t)y(t) dt = K(p) ˜y(p),
0
we obtain the following equation for the transform ˜y(p):
˜
˜
K(p) ˜y(p)= f(p). (3)
The solution of Eq. (3) is given by the formula
˜
f(p)
˜ y(p)= . (4)
˜
K(p)
On applying the Laplace inversion formula (if it is applicable) to (4), we obtain a solution of Eq. (1)
in the form c+i∞ ˜
1 f(p) px
y(x)= e dp. (5)
˜
2πi c–i∞ K(p)
When applying formula (5) in practice, the following two technical problems occur:

∞
˜
1 . Finding the transform K(p)= K(x)e –px dx for a given kernel K(x).
◦
0
˜
◦
2 . Finding the resolvent (5) whose transform R(p) is given by formula (4).
To calculate the corresponding integrals, tables of direct and inverse Laplace transforms can be
applied (see Supplements 4 and 5), and, in many cases, to find the inverse transform, methods of the
theory of functions of a complex variable are applied, including the Cauchy residue theorem (see
Subsection 7.1-4).
Remark. If the lower limit in the integral of a Volterra equation with difference kernel is a, then
this equation can be reduced to Eq. (1) by means of the change of variables x = ¯x – a, t = ¯ t – a.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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