In
mathematics, the Volterra integral equations are a special type of
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n ...
s. They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
:
where ''f'' is a given function and ''x'' is an unknown function to be solved for. A linear Volterra equation of the second kind is
:
In
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
, and in
Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...
, the corresponding operators are called
Volterra operator
In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space ''L''2 ,1of complex-valued square-integrable functions on the interval ,1 On the s ...
s. A useful method to solve such equations, the
Adomian decomposition method The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The method was developed from the 1970s to the 1990s by George Adomian, chair of the Center for Applied Mathema ...
, is due to
George Adomian
George Adomian (March 21, 1922 – June 17, 1996) was an American mathematician of Armenian descent who developed the Adomian decomposition method (ADM) for solving nonlinear differential equations, both ordinary and partial. The method is expla ...
.
A linear Volterra integral equation is a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
equation if
:
The function
in the integral is called the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine lea ...
. Such equations can be analyzed and solved by means of
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
techniques.
For a weakly singular kernel of the form
with
, Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation.
The Volterra integral equations were introduced by
Vito Volterra
Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis.
Biography
Born in An ...
and then studied by
Traian Lalescu
Traian Lalescu (; 12 July 1882 – 15 June 1929) was a Romanian mathematician. His main focus was on integral equations and he contributed to work in the areas of functional equations, trigonometric series, mathematical physics, geometry, mechani ...
in his 1908 thesis, ''Sur les équations de Volterra'', written under the direction of
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at th ...
. In 1911, Lalescu wrote the first book ever on integral equations.
Volterra integral equations find application in
demography
Demography () is the statistical study of populations, especially human beings.
Demographic analysis examines and measures the dimensions and dynamics of populations; it can cover whole societies or groups defined by criteria such as ed ...
as
Lotka's integral equation, the study of
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...
materials,
in
actuarial science through the
renewal equation
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) h ...
, and in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
to describe the flow behavior near finite-sized boundaries.
Conversion of Volterra equation of the first kind to the second kind
A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that
. Taking the derivative of the first kind Volterra equation gives us:
Dividing through by
yields:
Defining
and
completes the transformation of the first kind equation into a linear Volterra equation of the second kind.
Numerical solution using trapezoidal rule
A standard method for computing the numerical solution of a linear Volterra equation of the second kind is the
trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
\int_a^b f(x) \, dx.
The trapezoidal rule works b ...
, which for equally-spaced subintervals
is given by: