[1] In mathematical notation, integral equations may thus be expressed as being of the form:
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:
[1] Due to this close connection between differential and integral equations, one can often convert between the two.
[2] See also, for example, Green's function and Fredholm theory.
Certain kinds of nonlinear integral equations have specific names.
In addition, a linear Volterra integral equation of the second kind for an unknown function
Note that while throughout this article, the bounds of the integral are usually written as intervals, this need not be the case.
[7] In general, integral equations don't always need to be defined over an interval
[1] Another example of a singular integral equation in which the kernel becomes unbounded is:[1]
Strongly singular: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.
, the second-kind Volterra integral equation has a unique solution
is the resolvent kernel of K.[3] As defined above, a VFIE has the form:
satisfies the following conditions: Then the VFIE has a unique solution
A special type of Volterra equation which is used in various applications is defined as follows:[3]
[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] 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.
This gives a linear homogeneous Fredholm equation of the second type.
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.
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.
is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular.
In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:[3]
In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.
[3] Theorem — Suppose that the semi-linear Hammerstein equation has a unique solution
denotes the unique solution of the linear part of the equation above and is given by:
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.