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An [[Integral equation|integral equation]] of the form

$$ \tag{1 }
\int\limits _ { a } ^ { x }  K ( x, s) \phi ( s)  ds  =  f ( x)
$$

(a linear Volterra integral equation of the first kind), or of the form

$$ \tag{2 }
\phi ( x) - \lambda \int\limits _ { a } ^ { x }  K ( x, s) \phi ( s)  ds  =  f ( x)
$$

(a linear Volterra integral equation of the second kind). Here,  $  x, s, a $
are real numbers,  $  \lambda $
is a (generally complex) parameter,  $  \phi ( s) $
is an unknown function,  $  f( x) $, 
$  K( x, s) $
are given functions which are square-integrable on  $  [ a, b] $
and in the domain  $  a \leq  x \leq  b $, 
$  a \leq  s \leq  x $, 
respectively. The function  $  f( x) $
is called the free term, while the function  $  K( x, s) $
is called the kernel.

Volterra equations may be regarded as a special case of Fredholm equations (cf. [[Fredholm equation|Fredholm equation]]), with the kernel  $  K( x, s) $
defined on the square  $  a \leq  x \leq  b $, 
$  a \leq  s \leq  b $
and vanishing in the triangle  $  a \leq  x < s \leq  b $. 
A Volterra equation of the second kind without free term is called a homogeneous Volterra equation. The expression

$$ 
\int\limits _ { a } ^ { x }  K ( x, s) \phi ( s)  ds
$$

defines an integral operator acting in  $  L _ {2} $; 
it is known as the [[Volterra operator|Volterra operator]].

Equations of type (2) were first systematically studied by V. Volterra [[#References|[1]]], [[#References|[2]]]. A special case of a Volterra equation (1), the [[Abel integral equation|Abel integral equation]], was first studied by N.H. Abel. The principal result of the theory of Volterra equations of the second kind may be described as follows. For each complex  $  \lambda \neq \infty $
there exists a square-integrable solution of the Volterra equation of the second kind which is, moreover, unique. This solution may be obtained by successive approximation (cf. [[Sequential approximation, method of|Sequential approximation, method of]]), i.e. as the limit of a mean-square-convergent sequence:

$$ \tag{3 }
\phi _ {n + 1 }  ( x)  =  \lambda \int\limits _ { a } ^ { x }  K ( x, s) \phi _ {n} ( s)  ds + f ( x),
$$

where  $  \phi _ {0} $
is an arbitrary square-integrable function. In the case of a continuous kernel  $  K( x, s) $
and  $  f \in C ([ a, b]) $, 
this sequence converges uniformly on  $  [ a, b] $
to a unique continuous solution.

The following theorems apply to Volterra equations of the first kind. If  $  f( s) $
and  $  K( x, s) $
are differentiable,  $  K( x, x) \neq 0 $, 
$  x \in [ a, b] $, 
and if  $  K( x, x) $
and  $  K _ {x} ^ { \prime } ( x, s) $
are square-summable on  $  [ a, b] $
and on  $  a \leq  x \leq  b $, 
$  a \leq  s \leq  b $, 
respectively, a Volterra equation of the first kind is equivalent to the Volterra equation of the second kind obtained by differentiation of the Volterra equation of the first kind and having the form:

$$ 
\phi ( x) + \int\limits _ { a } ^ { x }  
\frac{K _ {x} ^ { \prime } ( x, s) }{K ( x, x) }
 \phi ( s)  ds  = \ 

\frac{f ^ { \prime } ( x) }{K ( x, x) }
 .
$$

If  $  K( x, x) = 0 $
at least at one point, the solution of the Volterra equation of the first kind must be more thoroughly investigated. If, on the other hand,  $  K( x, x) \equiv 0 $, 
the differentiation operation may be repeated under certain conditions. If the differentiation is impossible or does not result in a Volterra equation of the second kind, this equation of the first kind may be solved, for example, using a regularization algorithm (cf. [[Regularization|Regularization]]).

In practical applications of Volterra equations of the second kind it is very important that its solution be found at least approximately, e.g. by the method of successive approximation. However, other methods are usually more convenient, and one such method will now be described. Let  $  f $
and  $  K $
be continuous functions. The interval  $  [ a, b] $
is subdivided into  $  N $
equal parts with the aid of partitioning points  $  x _ {i} $, 
and  $  x _ {0} = a $, 
$  x _ {N} = b $. 
To find the approximate value of  $  \phi ( x _ {i} ) $, 
the integral over the interval is replaced by a quadrature sum, for example using the rectangle formula with nodes  $  x _ {0} \dots x _ {i-} 1 $:

$$ 
\int\limits _ { a } ^ { {x _ i} } K ( x _ {i} , s) \phi ( s)  ds  \approx \ 
\sum _ {j = 0 } ^ { {i }  - 1 } K ( x _ {i} , x _ {j} ) \phi ( x _ {j} )
{
\frac{b - a }{N}
 } .
$$

The approximate value of  $  \phi ( x _ {i} ) $
is then obtained using collocation:

$$ \tag{4 }
\phi ( x _ {i} )  =  \lambda {
\frac{b - a }{N}
 }
\sum _ {j = 0 } ^ { {i }  - 1 } K ( x _ {i} , x _ {j} )
\phi ( x _ {j} ) + f ( x _ {i} ),
$$

$$ 
\phi ( x _ {0} )  =  f ( a).
$$

The values of the approximate solution at the points on  $  [ a, b] $
situated between the partitioning points may be found, for example, from the relation:

$$ \tag{5 }
\phi ( x)  \simeq \ 
\lambda {
\frac{b - a }{N}
 } \sum _ {j = 1 } ^ { {i }  - 1 }
K ( x, x _ {j} ) \phi ( x _ {j} ) + f ( x),
$$

$$ 
x _ {j - 1 }   <  x  \leq   x _ {j} .
$$

For  $  N \rightarrow \infty $
this approximate solution converges uniformly to the exact solution of the Volterra equation of the second kind.

Many modifications of the above method are possible.

Everything said so far also applies to Volterra equations whose kernel  $  K( x, s) $
is a matrix of dimension  $  r \times r $, 
and where  $  \phi $
and  $  f $
are  $  r $-
dimensional vector-functions.

The name Volterra equation or generalized Volterra equation is also given to a more general integral equation, of the form:

$$ \tag{6 }
\phi ( P) - \lambda \int\limits _ {D ( P) }
K ( P, Q) \phi ( Q)  dQ  =  f ( P),
$$

if the successive approximations such as (3) are in some sense convergent (e.g. uniformly or on the average) on the domain of definition of the functions  $  \phi $
and  $  f $
for all  $  \lambda \neq \infty $. 
Here  $  P $
and  $  Q $
are points of the  $  n $-
dimensional Euclidean space,  $  D( P) $
is the domain of integration, which usually depends on the point  $  P $, 
and  $  D( P) \subseteq D $
for any  $  P $. 
The following equation may serve as an example:

$$ 
\phi ( x, y) - \lambda \int\limits _ { a } ^ { x }  \int\limits _ { a } ^ { b } 
K ( x, y, \xi , \eta ) \phi ( \xi , \eta )  d \xi  d \eta  =  f( x, y).
$$

If the function  $  K( x, y, \xi , \eta ) $
is square-integrable for  $  a \leq  x \leq  b $, 
$  a \leq  y \leq  b $, 
$  a \leq  \xi \leq  b $, 
$  a \leq  \eta \leq  b $, 
while  $  f( x, y) $
is square-integrable for  $  a \leq  x \leq  b $, 
$  a \leq  y \leq  b $, 
the sequence (3) is mean-square convergent for  $  \lambda \neq \infty $. 
Generalized Volterra equations of the first kind usually cannot be reduced to Volterra equations of the second kind, though this may be possible in special cases.

A further generalization of Volterra equations of types (2) and (6) is the linear operator equation:

$$ \tag{7 }
\phi - \lambda A \phi  =  f,
$$

where  $  \phi $
and  $  f $
are elements of a Banach space  $  E $, 
$  \lambda $
is a complex parameter and  $  A $
is a completely-continuous linear operator (cf. [[Completely-continuous operator|Completely-continuous operator]]). This equation is known as a Volterra operator equation, while the operator  $  A $
is known as a Volterra operator, or abstract Volterra operator, if the operator  $  ( I - \lambda A ) $
is invertible in  $  E $
for all  $  \lambda \neq \infty $. 
In such a case a sequence of the following type:  $  \phi _ {0} \in E $
is arbitrary,  $  \phi _ {n+} 1 = \lambda A \phi _ {n} + f $, 
converges in the norm of  $  E $
to a solution of equation (7). In the modern theory of Volterra operators and Volterra equations, deep relationships have been established between abstract and ordinary Volterra operators.

Non-linear Volterra equations is the name sometimes given to Volterra equations in which the product  $  K( x, s) \phi ( s) $
has been replaced by some function  $  K( x, s, \phi ( s)) $
which is non-linear with respect to  $  \phi ( s) $. 
Equations of this type are frequently encountered in theoretical and in applied studies. Thus, the [[Cauchy problem|Cauchy problem]] for an ordinary differential equation may be readily reduced to the problem of solving a non-linear Volterra equation. The application of potential theory to boundary value problems for equations of parabolic type reduces such problems to a generalized Volterra equation. In the case of non-linear Volterra equations it may be shown, if certain assumptions are made with respect to  $  K( x, s, \phi ( s) ) $, 
that successive approximations of type (3) converge on an interval  $  [ a, a + \Delta a] $, 
where  $  \Delta a $
is sufficiently small. Approximate solutions of non-linear Volterra equations are found by using the recurrence relation (4); it is sufficient to replace  $  K( x _ {i} , x _ {j} ) \phi ( x _ {j} ) $
by  $  K ( x _ {i} , x _ {j} , \phi ( x _ {j} )) $. 
If  $  K( x, s, \phi ( s) ) $
is independent of  $  x $, 
this method becomes identical with the [[Euler method|Euler method]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V. Volterra,   "Sulla inversione degli integrali definiti"  ''Rend. Accad. Lincei'' , '''5'''  (1896)  pp. 177–185; 289–300</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Volterra,   "Sopra alcune questioni di inversione di integrali definiti"  ''Ann. di Math. (2)'' , '''25'''  (1897)  pp. 139–187</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.G. Petrovskii,   "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.N. Tikhonov,   "Sur les équations fonctionnelles de Volterra et leurs applications à certains problèmes de la physique mathématique"  ''Byull. Moskov. Gos. Univ. (A)'' , '''1''' :  8  (1938)  pp. 1–25</TD></TR></table>

====Comments====
The numerical method given above is the special case of the Nyström method for Volterra equations. While for general Fredholm equations, (4) is a linear system to be solved, this system has the form of a recurrence relation here. For other numerical methods, see [[#References|[a1]]]. Volterra equations of the first kind are in general ill-posed (cf. [[Ill-posed problems|Ill-posed problems]]). If reduced to a second-kind equation by differentiation, this ill-posedness is contained in the differentiation of  $  f $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.T.H. Baker,   "The numerical treatment of integral equations" , Clarendon Press  (1977)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.A. Burton,   "Volterrra integral and differential equations" , Acad. Press  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.K. Miller,   "Nonlinear Volterra integral equations" , Benjamin  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Corduneanu,   "Integral equations and applications" , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   M.A. Kaashoek,   "Classes of linear operators" , '''1''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,   M.G. Krein,   "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  K. Jörgens,   "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.E. Taylor,   D.C. Lay,   "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>