Power series solution for integral equations

 

Power series solution for integral equations[edit]

See also: Liouville–Neumann series

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

${\displaystyle �(�)=�{\int }_0^{\mathrm{\infty }}�(��)�(�)��}$

in the form of a power series

${\displaystyle �(�)=\sum _{�=0}^{\mathrm{\infty }}\frac{{�}_{�}}{�(�+1)}{�}^{�}}$

where

${\displaystyle �(�)=\sum _{�=0}^{\mathrm{\infty }}{�}_{�}{�}^{-�},�(�+1)={\int }_0^{\mathrm{\infty }}�(�){�}^{�}��}$

are the Z-transform of the function g(s), and M(n + 1) is the Mellin transform of the Kernel.

Numerical solution[edit]

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

${\displaystyle \sum _{�=1}^{�}{�}_{�}�\left({�}_{�},{�}_{�}\right)�({�}_{�})=�({�}_{�}),�=0,1,\dots ,�.}$

Then we have a system with n equations and n variables. By solving it we get the value of the n variables

${\displaystyle �({�}_0),�({�}_1),\dots ,�({�}_{�}).}$

Integral equations as a generalization of eigenvalue equations[edit]

Further information: Fredholm theory

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

${\displaystyle \sum _{�}{�}_{�,�}{�}_{�}=�{�}_{�}}$

where M = [M_i,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

${\displaystyle \int �(�,�)�(�)��=��(�),}$

where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction φ(y).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.

Wiener–Hopf integral equations[edit]

Main article: Wiener–Hopf method

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[edit]

A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:^[3]

$${\displaystyle �(�)={\int }_0^{�}�(�,�)�(�,�(�))��.}$$

Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:^[3]

$${\displaystyle �(�,�(�))={�}_1(�)-{\int }_0^{�}{�}_1(�,�)�(�,�(�))��}$$

where:

$${\displaystyle {�}_1(�):=\frac{{�}^{\prime }(�)}{�(�,�)}\text{and}{�}_1(�,�):=-\frac{1}{�(�,�)}\frac{\mathrm{\partial }�(�,�)}{\mathrm{\partial }�}.}$$

The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:^[3]

$${\displaystyle (\mathcal{�}�)(�):={\int }_0^{�}�(�,�)�(�,�(�))��}$$

Here ${\displaystyle �:�\times \mathbb{�}\to \mathbb{�}}$ is a smooth function while the kernel K may be continuous, i.e. bounded, or weakly singular.^[3] 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:^[3]

$${\displaystyle �(�)=�(�)+(\mathcal{�}�)(�)}$$

In certain applications, the nonlinearity of the function G may be treated as being only semi-linear in the form of:^[3]

$${\displaystyle �(�,�)=�+�(�,�)}$$

In this case, we the following semi-linear Volterra integral equation:^[3]

$${\displaystyle �(�)=�(�)+(\mathcal{�}�)(�)=�(�)+{\int }_0^{�}�(�,�)[�(�)+�(�,�(�))]��}$$

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 ${\displaystyle �\in �(�)}$ and ${\displaystyle �:�\times \mathbb{�}\to \mathbb{�}}$ be a Lipschitz continuous function. Then the solution of this eqution may be written in the form: ${\displaystyle �(�)={�}_{�}(�)+{\int }_0^{�}�(�,�)�(�,�(�))��}$ where ${\displaystyle {�}_{�}(�)}$ denotes the unique solution of the linear part of the equation above and is given by: ${\displaystyle {�}_{�}(�)=�(�)+{\int }_0^{�}�(�,�)�(�)��}$ with ${\displaystyle �(�,�)}$ denoting the resolvent kernel.

We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, ${\displaystyle \mathcal{�}}$ defined as follows:^[3]

$${\displaystyle (\mathcal{�}�)(�):=�(�,�(�))}$$

More about this can be found on page 75 of this book.^[3]

Applications[edit]

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.

• Actuarial science (ruin theory^[8])
• Computational electromagnetics
  • Boundary element method
• Inverse problems
  • Marchenko equation (inverse scattering transform)
• Options pricing under jump-diffusion^[9]
• Radiative transfer
• Viscoelasticity
• Fluid mechanics