<s>
In	O
mathematics	O
,	O
the	O
Volterra	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
are	O
a	O
special	O
type	O
of	O
integral	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
function	O
in	O
the	O
integral	O
is	O
called	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
For	O
a	O
weakly	O
singular	O
kernel	B-Algorithm
of	O
the	O
form	O
with	O
,	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
first	I-Algorithm
kind	I-Algorithm
can	O
conveniently	O
be	O
transformed	O
into	O
a	O
classical	O
Abel	O
integral	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
The	O
Volterra	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
were	O
introduced	O
by	O
Vito	O
Volterra	O
and	O
then	O
studied	O
by	O
Traian	O
Lalescu	O
in	O
his	O
1908	O
thesis	O
,	O
Sur	O
les	O
équations	O
de	O
Volterra	O
,	O
written	O
under	O
the	O
direction	O
of	O
Émile	O
Picard	O
.	O
</s>
<s>
In	O
1911	O
,	O
Lalescu	O
wrote	O
the	O
first	O
book	O
ever	O
on	O
integral	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Volterra	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
find	O
application	O
in	O
demography	O
as	O
Lotka	O
's	O
integral	B-Algorithm
equation	I-Algorithm
,	O
the	O
study	O
of	O
viscoelastic	O
materials	O
,	O
</s>
<s>
A	O
linear	O
Volterra	B-Algorithm
equation	I-Algorithm
of	O
the	O
first	O
kind	O
can	O
always	O
be	O
reduced	O
to	O
a	O
linear	O
Volterra	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
,	O
assuming	O
that	O
.	O
</s>
<s>
Taking	O
the	O
derivative	O
of	O
the	O
first	O
kind	O
Volterra	B-Algorithm
equation	I-Algorithm
gives	O
us:Dividing	O
through	O
by	O
yields:Defining	O
and	O
completes	O
the	O
transformation	O
of	O
the	O
first	O
kind	O
equation	O
into	O
a	O
linear	O
Volterra	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
.	O
</s>
<s>
A	O
standard	O
method	O
for	O
computing	O
the	O
numerical	O
solution	O
of	O
a	O
linear	O
Volterra	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
is	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
which	O
for	O
equally-spaced	O
subintervals	O
is	O
given	O
by:Assuming	O
equal	O
spacing	O
for	O
the	O
subintervals	O
,	O
the	O
integral	O
component	O
of	O
the	O
Volterra	B-Algorithm
equation	I-Algorithm
may	O
be	O
approximated	O
by:Defining	O
,	O
,	O
and	O
,	O
we	O
have	O
the	O
system	O
of	O
linear	O
equations:This	O
is	O
equivalent	O
to	O
the	O
matrix	O
equation:For	O
well-behaved	O
kernels	B-Algorithm
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
tends	O
to	O
work	O
well	O
.	O
</s>
<s>
One	O
area	O
where	O
Volterra	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
appear	O
is	O
in	O
ruin	O
theory	O
,	O
the	O
study	O
of	O
the	O
risk	O
of	O
insolvency	O
in	O
actuarial	O
science	O
.	O
</s>
<s>
Under	O
these	O
circumstances	O
,	O
the	O
ruin	O
probability	O
may	O
be	O
represented	O
by	O
a	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	O
the	O
form:where	O
is	O
the	O
survival	O
function	O
of	O
the	O
claims	O
distribution	O
.	O
</s>
