<s>
In	O
mathematics	O
,	O
integral	B-Algorithm
equations	I-Algorithm
are	O
equations	O
in	O
which	O
an	O
unknown	O
function	O
appears	O
under	O
an	O
integral	O
sign	O
.	O
</s>
<s>
In	O
mathematical	O
notation	O
,	O
integral	B-Algorithm
equations	I-Algorithm
may	O
thus	O
be	O
expressed	O
as	O
being	O
of	O
the	O
form	O
:	O
where	O
is	O
an	O
integral	O
operator	O
acting	O
on	O
u	O
.	O
</s>
<s>
Hence	O
,	O
integral	B-Algorithm
equations	I-Algorithm
may	O
be	O
viewed	O
as	O
the	O
analog	O
to	O
differential	O
equations	O
where	O
instead	O
of	O
the	O
equation	O
involving	O
derivatives	O
,	O
the	O
equation	O
contains	O
integrals	O
.	O
</s>
<s>
A	O
direct	O
comparison	O
can	O
be	O
seen	O
with	O
the	O
mathematical	O
form	O
of	O
the	O
general	O
integral	B-Algorithm
equation	I-Algorithm
above	O
with	O
the	O
general	O
form	O
of	O
a	O
differential	O
equation	O
which	O
may	O
be	O
expressed	O
as	O
follows:where	O
may	O
be	O
viewed	O
as	O
a	O
differential	O
operator	O
of	O
order	O
i	O
.	O
</s>
<s>
Due	O
to	O
this	O
close	O
connection	O
between	O
differential	O
and	O
integral	B-Algorithm
equations	I-Algorithm
,	O
one	O
can	O
often	O
convert	O
between	O
the	O
two	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
method	O
of	O
solving	O
a	O
boundary	O
value	O
problem	O
is	O
by	O
converting	O
the	O
differential	O
equation	O
with	O
its	O
boundary	O
conditions	O
into	O
an	O
integral	B-Algorithm
equation	I-Algorithm
and	O
solving	O
the	O
integral	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Various	O
classification	O
methods	O
for	O
integral	B-Algorithm
equations	I-Algorithm
exist	O
.	O
</s>
<s>
A	O
few	O
standard	O
classifications	O
include	O
distinctions	O
between	O
linear	O
and	O
nonlinear	O
;	O
homogenous	O
and	O
inhomogenous	O
;	O
Fredholm	O
and	O
Volterra	O
;	O
first	O
order	O
,	O
second	O
order	O
,	O
and	O
third	O
order	O
;	O
and	O
singular	O
and	O
regular	O
integral	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
linear	O
if	O
the	O
unknown	O
function	O
u(x )	O
and	O
its	O
integrals	O
appear	O
linear	O
in	O
the	O
equation	O
.	O
</s>
<s>
Hence	O
,	O
an	O
example	O
of	O
a	O
linear	O
equation	O
would	O
be:As	O
a	O
note	O
on	O
naming	O
convention	O
:	O
i	O
)	O
u(x )	O
is	O
called	O
the	O
unknown	O
function	O
,	O
ii	O
)	O
f(x )	O
is	O
called	O
a	O
known	O
function	O
,	O
iii	O
)	O
K(x,t )	O
is	O
a	O
function	O
of	O
two	O
variables	O
and	O
often	O
called	O
the	O
Kernel	B-Algorithm
function	O
,	O
and	O
iv	O
)	O
λ	O
is	O
an	O
unknown	O
factor	O
or	O
parameter	O
,	O
which	O
plays	O
the	O
same	O
role	O
as	O
the	O
eigenvalue	O
in	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
nonlinear	O
if	O
the	O
unknown	O
function	O
u(x )	O
or	O
any	O
of	O
its	O
integrals	O
appear	O
nonlinear	O
in	O
the	O
equation	O
.	O
</s>
<s>
Hence	O
,	O
examples	O
of	O
nonlinear	O
equations	O
would	O
be	O
the	O
equation	O
above	O
if	O
we	O
replaced	O
u(t )	O
with	O
,	O
such	O
as:Certain	O
kinds	O
of	O
nonlinear	O
integral	B-Algorithm
equations	I-Algorithm
have	O
specific	O
names	O
.	O
</s>
<s>
Nonlinear	O
Volterra	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
of	O
the	O
second	O
kind	O
which	O
have	O
the	O
general	O
form	O
:	O
where	O
is	O
a	O
known	O
function	O
.	O
</s>
<s>
Nonlinear	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
of	O
the	O
second	O
kind	O
which	O
have	O
the	O
general	O
form	O
:	O
.	O
</s>
<s>
A	O
special	O
type	O
of	O
nonlinear	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
of	O
the	O
second	O
kind	O
are	O
given	O
by	O
the	O
form	O
:	O
,	O
which	O
has	O
the	O
two	O
special	O
subclasses	O
:	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
an	O
integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
first	O
kind	O
if	O
the	O
unknown	O
function	O
appears	O
only	O
under	O
the	O
integral	O
sign	O
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
an	O
integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
if	O
the	O
unknown	O
function	O
appears	O
also	O
outside	O
the	O
integral	O
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
an	O
integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
third	O
kind	O
if	O
it	O
is	O
a	O
linear	O
Integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
following	O
form:where	O
g(t )	O
vanishes	O
at	O
least	O
once	O
in	O
the	O
interval	O
 [ a , b ] 	O
or	O
where	O
g(t )	O
vanishes	O
at	O
a	O
finite	O
number	O
of	O
points	O
in	O
(	O
a	O
,	O
b	O
)	O
.	O
</s>
<s>
Fredholm	O
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
a	O
Fredholm	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
if	O
both	O
of	O
the	O
limits	O
of	O
integration	O
in	O
all	O
integrals	O
are	O
fixed	O
and	O
constant	O
.	O
</s>
<s>
Hence	O
,	O
the	O
following	O
two	O
examples	O
are	O
Fredholm	B-Algorithm
equations	I-Algorithm
:	O
</s>
<s>
Fredholm	B-Algorithm
equation	I-Algorithm
of	O
the	O
first	O
type	O
:	O
.	O
</s>
<s>
Fredholm	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
type	O
:	O
</s>
<s>
Note	O
that	O
we	O
can	O
express	O
integral	B-Algorithm
equations	I-Algorithm
such	O
as	O
those	O
above	O
also	O
using	O
integral	O
operator	O
notation	O
.	O
</s>
<s>
For	O
example	O
,	O
we	O
can	O
define	O
the	O
Fredholm	B-Algorithm
integral	I-Algorithm
operator	I-Algorithm
as:Hence	O
,	O
the	O
above	O
Fredholm	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
may	O
be	O
written	O
compactly	O
as	O
:	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
a	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
if	O
at	O
least	O
one	O
of	O
the	O
limits	O
of	O
integration	O
is	O
a	O
variable	O
.	O
</s>
<s>
Examples	O
of	O
Volterra	B-Algorithm
equations	I-Algorithm
would	O
be	O
:	O
</s>
<s>
Volterrra	O
integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
first	O
kind	O
:	O
</s>
<s>
Volterrra	O
integral	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
kind	O
:	O
</s>
<s>
As	O
with	O
Fredholm	B-Algorithm
equations	I-Algorithm
,	O
we	O
can	O
again	O
adopt	O
operator	O
notation	O
.	O
</s>
<s>
Thus	O
,	O
we	O
can	O
define	O
the	O
linear	O
Volterra	O
integral	O
operator	O
,	O
as	O
follows:where	O
and	O
K(t,s )	O
is	O
called	O
the	O
kernel	B-Algorithm
and	O
must	O
be	O
continuous	O
on	O
the	O
interval	O
.	O
</s>
<s>
Hence	O
,	O
the	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
first	I-Algorithm
kind	I-Algorithm
may	O
be	O
written	O
as:with	O
.	O
</s>
<s>
In	O
addition	O
,	O
a	O
linear	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
kind	I-Algorithm
for	O
an	O
unknown	O
function	O
and	O
a	O
given	O
continuous	O
function	O
on	O
the	O
interval	O
where	O
::	O
In	O
higher	O
dimensions	O
,	O
integral	B-Algorithm
equations	I-Algorithm
such	O
as	O
Fredholm-Volterra	O
integral	B-Algorithm
equations	I-Algorithm
(	O
VFIE	O
)	O
exist	O
.	O
</s>
<s>
In	O
general	O
,	O
integral	B-Algorithm
equations	I-Algorithm
do	O
n't	O
always	O
need	O
to	O
be	O
defined	O
over	O
an	O
interval	O
,	O
but	O
could	O
also	O
be	O
defined	O
over	O
a	O
curve	O
or	O
surface	O
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
homogeneous	O
if	O
the	O
known	O
function	O
is	O
identically	O
zero	O
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
homogeneous	O
if	O
the	O
known	O
function	O
is	O
nonzero	O
.	O
</s>
<s>
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
regular	O
if	O
the	O
integrals	O
used	O
are	O
all	O
proper	O
integrals	O
.	O
</s>
<s>
or	O
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
singular	O
or	O
weakly	O
singular	O
if	O
the	O
integral	O
is	O
an	O
improper	O
integral	O
.	O
</s>
<s>
This	O
could	O
be	O
either	O
because	O
at	O
least	O
one	O
of	O
the	O
limits	O
of	O
integration	O
is	O
infinite	O
or	O
the	O
kernel	B-Algorithm
becomes	O
unbounded	O
,	O
meaning	O
infinite	O
,	O
on	O
at	O
least	O
one	O
point	O
in	O
the	O
interval	O
or	O
domain	O
over	O
which	O
is	O
being	O
integrated	O
.	O
</s>
<s>
Examples	O
include:These	O
two	O
integral	B-Algorithm
equations	I-Algorithm
are	O
the	O
Fourier	O
transform	O
and	O
the	O
Laplace	O
transform	O
of	O
u(x )	O
,	O
respectively	O
,	O
with	O
both	O
being	O
Fredholm	B-Algorithm
equations	I-Algorithm
of	O
the	O
first	O
kind	O
with	O
kernel	B-Algorithm
and	O
,	O
respectively	O
.	O
</s>
<s>
Another	O
example	O
of	O
a	O
singular	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
in	O
which	O
the	O
kernel	B-Algorithm
becomes	O
unbounded	O
is	O
:	O
This	O
equation	O
is	O
a	O
special	O
form	O
of	O
the	O
more	O
general	O
weakly	O
singular	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
first	I-Algorithm
kind	I-Algorithm
,	O
called	O
Abel	O
's	O
integral	B-Algorithm
equation	I-Algorithm
:	O
:	O
An	O
integral	B-Algorithm
equation	I-Algorithm
is	O
called	O
strongly	O
singular	O
if	O
the	O
integral	O
is	O
defined	O
by	O
a	O
special	O
regularisation	O
,	O
for	O
example	O
,	O
by	O
the	O
Cauchy	O
principal	O
value	O
.	O
</s>
<s>
An	O
Integro-differential	O
equation	O
,	O
as	O
the	O
name	O
suggests	O
,	O
combines	O
differential	O
and	O
integral	B-Algorithm
operators	I-Algorithm
into	O
one	O
equation	O
.	O
</s>
<s>
The	O
solution	O
to	O
a	O
linear	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
first	I-Algorithm
kind	I-Algorithm
,	O
given	O
by	O
the	O
equation:can	O
be	O
described	O
by	O
the	O
following	O
uniqueness	O
and	O
existence	O
theorem	O
.	O
</s>
<s>
Recall	O
that	O
the	O
Volterra	O
integral	O
operator	O
,	O
can	O
be	O
defined	O
as	O
follows:where	O
and	O
K(t,s )	O
is	O
called	O
the	O
kernel	B-Algorithm
and	O
must	O
be	O
continuous	O
on	O
the	O
interval	O
.	O
</s>
<s>
The	O
solution	O
to	O
a	O
linear	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
kind	I-Algorithm
,	O
given	O
by	O
the	O
equation:can	O
be	O
described	O
by	O
the	O
following	O
uniqueness	O
and	O
existence	O
theorem	O
.	O
</s>
<s>
A	O
Volterra	B-Algorithm
Integral	I-Algorithm
equation	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
kind	I-Algorithm
can	O
be	O
expressed	O
as	O
follows:where	O
,	O
,	O
and	O
.	O
</s>
<s>
This	O
integral	B-Algorithm
equation	I-Algorithm
has	O
a	O
unique	O
solution	O
given	O
by:where	O
is	O
the	O
resolvent	O
kernel	B-Algorithm
of	O
K	O
.	O
</s>
<s>
The	O
Fredholm-Volterrra	O
Integral	O
Operator	O
is	O
defined	O
as:In	O
the	O
case	O
where	O
the	O
Kernel	B-Algorithm
K	O
may	O
be	O
written	O
as	O
,	O
K	O
is	O
called	O
the	O
positive	O
memory	O
kernel	B-Algorithm
.	O
</s>
<s>
A	O
special	O
type	O
of	O
Volterra	B-Algorithm
equation	I-Algorithm
which	O
is	O
used	O
in	O
various	O
applications	O
is	O
defined	O
as	O
follows:where	O
,	O
the	O
function	O
g(t )	O
is	O
continuous	O
on	O
the	O
interval	O
,	O
and	O
the	O
Volterra	O
integral	O
operator	O
is	O
given	O
by:with	O
.	O
</s>
<s>
In	O
the	O
following	O
section	O
,	O
we	O
give	O
an	O
example	O
of	O
how	O
to	O
convert	O
an	O
initial	O
value	O
problem	O
(	O
IVP	O
)	O
into	O
an	O
integral	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
There	O
are	O
multiple	O
motivations	O
for	O
doing	O
so	O
,	O
among	O
them	O
being	O
that	O
integral	B-Algorithm
equations	I-Algorithm
can	O
often	O
be	O
more	O
readily	O
solvable	O
and	O
are	O
more	O
suitable	O
for	O
proving	O
existence	O
and	O
uniqueness	O
theorems	O
.	O
</s>
<s>
Rearranging	O
the	O
equation	O
above	O
,	O
we	O
get	O
the	O
integral	B-Algorithm
equation	I-Algorithm
:	O
</s>
<s>
which	O
is	O
a	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	O
the	O
form	O
:	O
</s>
<s>
where	O
K(x,t )	O
is	O
called	O
the	O
kernel	B-Algorithm
and	O
equal	O
to	O
2t	O
,	O
and	O
f(x )	O
=	O
1	O
.	O
</s>
<s>
are	O
the	O
-transform	O
of	O
the	O
function	O
,	O
and	O
is	O
the	O
Mellin	O
transform	O
of	O
the	O
Kernel	B-Algorithm
.	O
</s>
<s>
It	O
is	O
worth	O
noting	O
that	O
integral	B-Algorithm
equations	I-Algorithm
often	O
do	O
not	O
have	O
an	O
analytical	O
solution	O
,	O
and	O
must	O
be	O
solved	O
numerically	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
evaluating	O
the	O
electric-field	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
(	O
EFIE	B-Algorithm
)	O
or	O
magnetic-field	O
integral	B-Algorithm
equation	I-Algorithm
(	O
MFIE	O
)	O
over	O
an	O
arbitrarily	O
shaped	O
object	O
in	O
an	O
electromagnetic	O
scattering	O
problem	O
.	O
</s>
<s>
Certain	O
homogeneous	O
linear	O
integral	B-Algorithm
equations	I-Algorithm
can	O
be	O
viewed	O
as	O
the	O
continuum	B-Algorithm
limit	I-Algorithm
of	O
eigenvalue	O
equations	O
.	O
</s>
<s>
where	O
the	O
sum	O
over	O
has	O
been	O
replaced	O
by	O
an	O
integral	O
over	O
and	O
the	O
matrix	O
and	O
the	O
vector	O
have	O
been	O
replaced	O
by	O
the	O
kernel	B-Algorithm
and	O
the	O
eigenfunction	B-Algorithm
.	O
</s>
<s>
This	O
gives	O
a	O
linear	O
homogeneous	O
Fredholm	B-Algorithm
equation	I-Algorithm
of	O
the	O
second	O
type	O
.	O
</s>
<s>
If	O
the	O
distribution	O
has	O
support	O
only	O
at	O
the	O
point	O
,	O
then	O
the	O
integral	B-Algorithm
equation	I-Algorithm
reduces	O
to	O
a	O
differential	B-Algorithm
eigenfunction	I-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
Volterra	O
and	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
can	O
arise	O
from	O
a	O
single	O
differential	O
equation	O
,	O
depending	O
on	O
which	O
sort	O
of	O
conditions	O
are	O
applied	O
at	O
the	O
boundary	O
of	O
the	O
domain	O
of	O
its	O
solution	O
.	O
</s>
<s>
Originally	O
,	O
such	O
equations	O
were	O
studied	O
in	O
connection	O
with	O
problems	O
in	O
radiative	O
transfer	O
,	O
and	O
more	O
recently	O
,	O
they	O
have	O
been	O
related	O
to	O
the	O
solution	O
of	O
boundary	O
integral	B-Algorithm
equations	I-Algorithm
for	O
planar	O
problems	O
in	O
which	O
the	O
boundary	O
is	O
only	O
piecewise	O
smooth	O
.	O
</s>
<s>
A	O
Hammerstein	O
equation	O
is	O
a	O
nonlinear	O
first-kind	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	O
the	O
form:Under	O
certain	O
regularity	O
conditions	O
,	O
the	O
equation	O
is	O
equivalent	O
to	O
the	O
implicit	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
of	O
the	O
second-kind:where:The	O
equation	O
may	O
however	O
also	O
be	O
expressed	O
in	O
operator	O
form	O
which	O
motivates	O
the	O
definition	O
of	O
the	O
following	O
operator	O
called	O
the	O
nonlinear	O
Volterra-Hammerstein	O
operator:Here	O
is	O
a	O
smooth	O
function	O
while	O
the	O
kernel	B-Algorithm
K	O
may	O
be	O
continuous	O
,	O
i.e.	O
</s>
<s>
The	O
corresponding	O
second-kind	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
called	O
the	O
Volterra-Hammerstein	O
Integral	B-Algorithm
Equation	I-Algorithm
of	O
the	O
second	O
kind	O
,	O
or	O
simply	O
Hammerstein	O
equation	O
for	O
short	O
,	O
can	O
be	O
expressed	O
as:In	O
certain	O
applications	O
,	O
the	O
nonlinearity	O
of	O
the	O
function	O
G	O
may	O
be	O
treated	O
as	O
being	O
only	O
semi-linear	O
in	O
the	O
form	O
of:In	O
this	O
case	O
,	O
we	O
the	O
following	O
semi-linear	O
Volterra	O
integral	O
equation:In	O
this	O
form	O
,	O
we	O
can	O
state	O
an	O
existence	O
and	O
uniqueness	O
theorem	O
for	O
the	O
semi-linear	O
Hammerstein	O
integral	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Integral	B-Algorithm
equations	I-Algorithm
are	O
important	O
in	O
many	O
applications	O
.	O
</s>
<s>
Problems	O
in	O
which	O
integral	B-Algorithm
equations	I-Algorithm
are	O
encountered	O
include	O
radiative	O
transfer	O
,	O
and	O
the	O
oscillation	B-Algorithm
of	O
a	O
string	O
,	O
membrane	O
,	O
or	O
axle	O
.	O
</s>
<s>
Oscillation	B-Algorithm
problems	O
may	O
also	O
be	O
solved	O
as	O
differential	O
equations	O
.	O
</s>
