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In [[mathematics]], the '''Volterra integral equations''' are a special type of [[integral equation]]s. They are divided into two groups referred to as the first and the second kind.
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A linear Volterra equation of the first kind is
 
:<math> f(t) = \int_a^t K(t,s)\,x(s)\,ds</math>
 
where ''&fnof;'' is a given function and ''x'' is an unknown function to be solved for.  A linear Volterra equation of the second kind is
 
:<math> x(t) = f(t) + \int_a^t K(t,s)x(s)\,ds. </math>
 
In [[operator theory]], and in [[Fredholm theory]], the corresponding equations are called the [[Volterra operator]].
 
A linear Volterra integral equation is a [[convolution]] equation if
 
:<math> x(t) = f(t) + \int_{t_0}^t K(t-s)x(s)\,ds. </math>
 
The function <math> K </math> in the integral is often called the [[kernel (linear algebra)|kernel]].
Such equations can be analysed and solved by means of [[Laplace transform]] techniques.  
 
The Volterra integral equations were introduced by [[Vito Volterra]] and then studied by [[Traian Lalescu]] in his 1908 thesis, ''Sur les équations de Volterra'', written under the direction of [[Charles Émile Picard|Émile Picard]].   In 1911, Lalescu wrote the first book ever on integral equations.
 
Volterra integral equations find application in [[demography]], the study of [[viscoelastic]] materials,
and in [[Actuarial mathematics|insurance mathematics]] through the [[renewal equation]].
 
==References==
*Traian Lalescu, ''Introduction à la théorie des équations intégrales. Avec une préface de É. Picard'', [[Paris]]: [[Hermann (publishing house)|A. Hermann et Fils]], 1912. VII + 152 pp.
*{{springer|title=Volterra equation|id=p/v096840}}
*{{MathWorld|title=Volterra Integral Equation of the First Kind
|urlname=VolterraIntegralEquationoftheFirstKind}}
*{{MathWorld|title=Volterra Integral Equation of the Second Kind
|urlname=VolterraIntegralEquationoftheSecondKind}}
*[http://eqworld.ipmnet.ru/en/solutions/ie.htm Integral Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 19.2. Volterra Equations | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=992}}
 
[[Category:Integral equations]]
[[Category:Mathematical analysis]]

Revision as of 21:37, 3 March 2014

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