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

The Volterra equation of the second kind

x

y(x) – K(x, t)y(t) dt = f(x) (1)
a
can be reduced to a Volterra equation of the first kind in two ways.

9.8-1. The First Method
We integrate Eq. (1) with respect to x from a to x and then reverse the order of integration in the
double integral. We finally obtain the Volterra equation of the first kind
x

M(x, t)y(t) dt = F(x), (2)
a

where M(x, t) and F(x) are defined as follows:
x x
M(x, t)=1 – K(s, t) ds, F(x)= f(t) dt. (3)
t a

9.8-2. The Second Method

Assume that the condition f(a) = 0 is satisfied. In this case Eq. (1) can be reduced to a Volterra
equation of the first kind for the derivative of the unknown function,

x
N(x, t)y (t) dt = f(x), y(a)=0, (4)

t
a
where
x

N(x, t)=1 – K(x, s) ds. (5)
t
Indeed, on integrating by parts the right-hand side of formula (4) with regard to formula (5), we
arrive at Eq. (1).
Remark. For f(a) ≠ 0, Eq. (1) implies the relation y(a)= f(a). In this case the substitution
z(x)= y(x) – f(a) yields the Volterra equation of the second kind
x

z(x) – K(x, t)z(t) dt = Φ(x),
a
x

Φ(x)= f(x) – f(a)+ f(a) K(x, t) dt,
a
whose right-hand side satisfies the condition Φ(a) = 0, and hence this equation can be reduced by
the second method to a Volterra equation of the first kind.

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References for Section 9.8: V. Volterra (1959), A. F. Verlan’ and V. S. Sizikov (1986).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 485