Page 462 - Handbook Of Integral Equations

On differentiating formulas (10) and eliminating y(x) from the resulting equations, we arrive at the
following linear differential equations for the functions w m = w m (x):

h 1 (x)w m = h m (x)w , m =2, ... , n, (12)

1
(the prime stands for the derivative with respect to x) with the initial conditions

w m (a)=0, m =1, ... , n.

Any solution of system (11), (12) determines a solution of the original integral equation (9) by each
of the expressions
w (x)

m
y(x)= , m =1, ... , n,
h m (x)
which can be obtained by differentiating formula (10).
System (11), (12) can be reduced to a linear differential equation of order n – 1 for any
function w m (x)(m =1, ... , n) by multiple differentiation of Eq. (11) with regard to (12).

•
References for Section 8.2: E. Goursat (1923), A. F. Verlan’ and V. S. Sizikov (1986).

8.3. Reduction of Volterra Equations of the First Kind to
Volterra Equations of the Second Kind

8.3-1. The First Method
Suppose that the kernel and the right-hand side of the equation

x
K(x, t)y(t) dt = f(x), (1)
a
have continuous derivatives with respect to x and that the condition K(x, x) /≡ 0 holds. In this case,
after differentiating relation (1) and dividing the resulting expression by K(x, x) we arrive at the
following Volterra equation of the second kind:

x K (x, t) f (x)

y(x)+ x y(t) dt = x . (2)
a K(x, x) K(x, x)
Equations of this type are considered in Chapter 9. If K(x, x) ≡ 0, then, on differentiating Eq. (1)

with respect to x twice and assuming that K (x, t)| t=x /≡ 0, we obtain the Volterra equation of the
x
second kind

x K (x, t) f (x)

y(x)+ xx y(t) dt = xx .

x
a K (x, t)| t=x K (x, t)| t=x
x
If K (x, x) ≡ 0, we can again apply differentiation, and so on. If the first m – 2 partial derivatives

x
of the kernel with respect to x are identically zero and the (m – 1)st derivative is nonzero, then the
m-fold differentiation of the original equation gives the following Volterra equation of the second
kind:
x K (m) (x, t) f (m) (x)
y(x)+ x y(t) dt = x .
(m–1) (m–1)
a K x (x, t)| t=x K x (x, t)| t=x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 444