Integral equation


In mathematical analysis, 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:
where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:where may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory.

Classification and overview

Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogeneous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation. These comments are made concrete through the following definitions and examples:

Linearity

  • Nonlinear Volterra integral equations of the second kind which have the general form: where is a known function.
  • Nonlinear Fredholm integral equations of the second kind which have the general form:.
  • A special type of nonlinear Fredholm integral equations of the second kind is given by the form:, which has the two special subclasses:
  • * Urysohn equation:.
  • * Hammerstein equation:.
More information on the Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below.

Location of the unknown equation

Limits of Integration

  • Fredholm equation of the first type:.
  • Fredholm equation of the second type:
Note that we can express integral equations such as those above also using integral operator notation. For example, we can define the Fredholm integral operator as:Hence, the above Fredholm equation of the second kind may be written compactly as:
As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator, as follows:where and K is called the kernel and must be continuous on the interval. Hence, the Volterra integral equation of the first kind may be written as:with. In addition, a linear Volterra integral equation of the second kind for an unknown function and a given continuous function on the interval where :: In higher dimensions, integral equations such as Fredholm–Volterra integral equations exist. A VFIE has the form:with and being a closed bounded region in with piecewise smooth boundary. The Fredholm-Volterra Integral Operator is defined as:
Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case. In general, integral equations don't always need to be defined over an interval, but could also be defined over a curve or surface.

Homogeneity

Regularity

or : An integral equation is called singular or weakly singular if the integral is an improper integral. 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.
Examples include:These two integral equations are the Fourier transform and the Laplace transform of u, respectively, with both being Fredholm equations of the first kind with kernel and, respectively. Another example of a singular integral equation in which the kernel becomes unbounded is: This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation: : An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.

Integro-differential equations

An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation. There are many version including the Volterra integro-differential equation and delay type equations as defined below. For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as:For delay problems, we can define the delay integral operator as:where the delay integro-differential equation may be expressed as:

Volterra integral equations

Uniqueness and existence theorems in 1D

The solution to a linear Volterra integral equation of the first kind, given by the equation:can be described by the following uniqueness and existence theorem. Recall that the Volterra integral operator, can be defined as follows:where and K is called the kernel and must be continuous on the interval.
The solution to a linear Volterra integral equation of the second kind, given by the equation:can be described by the following uniqueness and existence theorem.

Volterra integral equations in

A Volterra Integral equation of the second kind can be expressed as follows:where,, and. This integral equation has a unique solution given by:where is the resolvent kernel of K.

Uniqueness and existence theorems of Fredholm–Volterra equations

As defined above, a VFIE has the form:with and being a closed bounded region in with piecewise smooth boundary. The Fredholm–Volterrra Integral Operator is defined as:In the case where the Kernel K may be written as, K is called the positive memory kernel. With this in mind, we can now introduce the following theorem:

Special Volterra equations

A special type of Volterra equation which is used in various applications is defined as follows:where, the function g is continuous on the interval, and the Volterra integral operator is given by:with.

Converting IVP to integral equations

In the following section, we give an example of how to convert an initial value problem 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.
The following example was provided by Wazwaz on pages 1 and 2 in his book. We examine the IVP given by the equation:
and the initial condition:
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 is called the kernel and equal to 2t, and f = 1.

Numerical solution

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 or magnetic-field integral equation 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 equations and variables. By solving it we get the value of the variables

Integral equations as a generalization of eigenvalue equations

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 is a matrix, is one of its eigenvectors, and is the associated eigenvalue.
Taking the continuum limit, i.e., replacing the discrete indices and with continuous variables and, yields
where the sum over has been replaced by an integral over and the matrix and the vector have been replaced by the kernel and the eigenfunction. This gives a linear homogeneous Fredholm equation of the second type.
In general, can be a distribution, rather than a function in the strict sense. If the distribution has support only at the point, 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.

Wiener–Hopf integral equations

Originally, such equations were studied in connection with problems in radiative transfer, and more recently, they have been related to the solution of boundary integral equations for planar problems in which the boundary is only piecewise smooth.

Hammerstein equations

A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:where:The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:Here is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular. The corresponding second-kind Volterra integral equation called the Volterra-Hammerstein Integral Equation of the second kind, or simply Hammerstein equation for short, can be expressed as:In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:In this case, we the following semi-linear Volterra integral equation:In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.
We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, defined as follows:More about this can be found on page 75 of this book.