In mathematics, integral equations are equations in which an unknown function appears under an integral sign. [1] In mathematical notation, integral equations may thus be expressed as being of the form:
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. [1] These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation. [1] These comments are made concrete through the following definitions and examples:
Linear: An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation. [1] Hence, an example of a linear equation would be: [1]
Nonlinear: An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in the equation. [1] Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with , such as:
More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.
First kind: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign. [3] An example would be: . [3]
Second kind: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral. [3]
Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: [3]
Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant. [1] An example would be that the integral is taken over a fixed subset of . [3] Hence, the following two examples are Fredholm equations: [1]
Note that we can express integral equations such as those above also using integral operator notation. [7] For example, we can define the Fredholm integral operator as:
Volterra: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable. [1] Hence, the integral is taken over a domain varying with the variable of integration. [3] Examples of Volterra equations would be: [1]
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator , as follows: [3]
Homogenous: An integral equation is called homogeneous if the known function is identically zero. [1]
Inhomogenous: An integral equation is called inhomogeneous if the known function is nonzero. [1]
Regular: An integral equation is called regular if the integrals used are all proper integrals. [7]
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral. [7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated. [1]
Examples include: [1]
An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation. [1] There are many version including the Volterra integro-differential equation and delay type equations as defined below. [3] For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as: [3]
The solution to a linear Volterra integral equation of the first kind, given by the equation:
Theorem — Assume that satisfies and for some Then for any with the integral equation above has a unique solution in .
The solution to a linear Volterra integral equation of the second kind, given by the equation: [3]
Theorem — Let and let denote the resolvent Kernel associated with . Then, for any , the second-kind Volterra integral equation has a unique solution and this solution is given by: .
A Volterra Integral equation of the second kind can be expressed as follows: [3]
As defined above, a VFIE has the form:
Theorem — If the linear VFIE given by: with satisfies the following conditions:
Then the VFIE has a unique solution given by where is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel and solves the resolvent equations:
A special type of Volterra equation which is used in various applications is defined as follows: [3]
In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems. [7]
The following example was provided by Wazwaz on pages 1 and 2 in his book. [1] We examine the IVP given by the equation:
If we integrate both sides of the equation, we get:
and by the fundamental theorem of calculus, we obtain:
Rearranging the equation above, we get the integral equation:
which is a Volterra integral equation of the form:
where K(x,t) is called the kernel and equal to 2t, and f(x)=1. [1]
In many cases, if the Kernel of the integral equation is of the form K(xt) and the Mellin transform of K(t) exists, we can find the solution of the integral equation
in the form of a power series
where
are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.
It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.
One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule
Then we have a system with n equations and n variables. By solving it we get the value of the n variables
Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as
where M = [Mi,j is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue.
Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields
where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction φ(y). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.
In general, K(x, y) can be a distribution, rather than a function in the strict sense. If the distribution K has support only at the point x = y, then the integral equation reduces to a differential eigenfunction equation.
In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.
A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form: [3]
Theorem — Suppose that the semi-linear Hammerstein equation has a unique solution and be a Lipschitz continuous function. Then the solution of this equation may be written in the form: where denotes the unique solution of the linear part of the equation above and is given by: with denoting the resolvent kernel.
We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, defined as follows: [3]
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.
{{
cite book}}
: CS1 maint: multiple names: authors list (
link)
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. [1] In mathematical notation, integral equations may thus be expressed as being of the form:
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. [1] These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation. [1] These comments are made concrete through the following definitions and examples:
Linear: An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation. [1] Hence, an example of a linear equation would be: [1]
Nonlinear: An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in the equation. [1] Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with , such as:
More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.
First kind: An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign. [3] An example would be: . [3]
Second kind: An integral equation is called an integral equation of the second kind if the unknown function also appears outside the integral. [3]
Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: [3]
Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant. [1] An example would be that the integral is taken over a fixed subset of . [3] Hence, the following two examples are Fredholm equations: [1]
Note that we can express integral equations such as those above also using integral operator notation. [7] For example, we can define the Fredholm integral operator as:
Volterra: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable. [1] Hence, the integral is taken over a domain varying with the variable of integration. [3] Examples of Volterra equations would be: [1]
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator , as follows: [3]
Homogenous: An integral equation is called homogeneous if the known function is identically zero. [1]
Inhomogenous: An integral equation is called inhomogeneous if the known function is nonzero. [1]
Regular: An integral equation is called regular if the integrals used are all proper integrals. [7]
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral. [7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated. [1]
Examples include: [1]
An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation. [1] There are many version including the Volterra integro-differential equation and delay type equations as defined below. [3] For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as: [3]
The solution to a linear Volterra integral equation of the first kind, given by the equation:
Theorem — Assume that satisfies and for some Then for any with the integral equation above has a unique solution in .
The solution to a linear Volterra integral equation of the second kind, given by the equation: [3]
Theorem — Let and let denote the resolvent Kernel associated with . Then, for any , the second-kind Volterra integral equation has a unique solution and this solution is given by: .
A Volterra Integral equation of the second kind can be expressed as follows: [3]
As defined above, a VFIE has the form:
Theorem — If the linear VFIE given by: with satisfies the following conditions:
Then the VFIE has a unique solution given by where is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel and solves the resolvent equations:
A special type of Volterra equation which is used in various applications is defined as follows: [3]
In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems. [7]
The following example was provided by Wazwaz on pages 1 and 2 in his book. [1] We examine the IVP given by the equation:
If we integrate both sides of the equation, we get:
and by the fundamental theorem of calculus, we obtain:
Rearranging the equation above, we get the integral equation:
which is a Volterra integral equation of the form:
where K(x,t) is called the kernel and equal to 2t, and f(x)=1. [1]
In many cases, if the Kernel of the integral equation is of the form K(xt) and the Mellin transform of K(t) exists, we can find the solution of the integral equation
in the form of a power series
where
are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.
It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.
One method to solve numerically requires discretizing variables and replacing integral by a quadrature rule
Then we have a system with n equations and n variables. By solving it we get the value of the n variables
Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations. Using index notation, an eigenvalue equation can be written as
where M = [Mi,j is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue.
Taking the continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y, yields
where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction φ(y). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.
In general, K(x, y) can be a distribution, rather than a function in the strict sense. If the distribution K has support only at the point x = y, then the integral equation reduces to a differential eigenfunction equation.
In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at the boundary of the domain of its solution.
A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form: [3]
Theorem — Suppose that the semi-linear Hammerstein equation has a unique solution and be a Lipschitz continuous function. Then the solution of this equation may be written in the form: where denotes the unique solution of the linear part of the equation above and is given by: with denoting the resolvent kernel.
We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, defined as follows: [3]
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.
{{
cite book}}
: CS1 maint: multiple names: authors list (
link)