Integral equation

35 languages

• Article
• Talk

• Read
• Edit
• View history

From Wikipedia, the free encyclopedia

For equations of integer unknowns, see Diophantine equation.

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:

$${\displaystyle �({�}_1,{�}_2,{�}_3,...,{�}_{�};�({�}_1,{�}_2,{�}_3,...,{�}_{�});{�}^1(�),{�}^2(�),{�}^3(�),...,{�}^{�}(�))=0}$$

where ${\displaystyle {�}^{�}(�)}$ is an integral operator acting on u.^[1] Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.^[1] 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:

$${\displaystyle �({�}_1,{�}_2,{�}_3,...,{�}_{�};�({�}_1,{�}_2,{�}_3,...,{�}_{�});{�}^1(�),{�}^2(�),{�}^3(�),...,{�}^{�}(�))=0}$$

where ${\displaystyle {�}^{�}(�)}$ may be viewed as a differential operator of order i.^[1] Due to this close connection between differential and integral equations, one can often convert between the two.^[1] 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.^[1] 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.^[2] See also, for example, Green's function and Fredholm theory.

Classification and overview[edit]

Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogenous; 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:

Linearity[edit]

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]

$${\displaystyle �(�)=�(�)+{\int }_{�(�)}^{�(�)}�(�,�)\cdot �(�)��}$$

As a note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in linear algebra.^[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 ${\displaystyle {�}^2(�),���(�(�)),\text{or }{�}^{�(�)}}$, such as:

$${\displaystyle �(�)=�(�)+{\int }_{�(�)}^{�(�)}�(�,�)\cdot {�}^2(�)��}$$

Certain kinds of nonlinear integral equations have specific names.^[3] A selection of such equations are:^[3]

• Nonlinear Volterra integral equations of the second kind which have the general form: ${\displaystyle �(�)=�(�)+�{\int }_{�}^{�}�(�,�)�(�,�,�(�))��,}$ where F is a known function.^[3]
• Nonlinear Fredholm integral equations of the second kind which have the general form: ${\displaystyle �(�)=�(�,{\int }_{�}^{�}�(�,�,�(�),�(�))��)}$.^[3]
• A special type of nonlinear Fredholm integral equations of the second kind are given by the form: ${\displaystyle �(�)=�(�)+{\int }_{�}^{�}�(�,�,�(�),�(�))��}$, which has the two special subclasses:^[3]
  • Urysohn equation: ${\displaystyle �(�)=�(�)+{\int }_{�}^{�}�(�,�,�(�))��}$.^[3]
  • Hammerstein equation: ${\displaystyle �(�)=�(�)+{\int }_{�}^{�}�(�,�)�(�,�(�))��}$.^[3]

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[edit]

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: ${\displaystyle �(�)={\int }_{�}^{�}�(�,�)�(�)��}$.^[3]

Second kind: An integral equation is called an integral equation of the second kind if the unknown function appears also 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]

$${\displaystyle �(�)�(�)+�{\int }_{�}^{�}�(�,�)�(�)��=�(�)}$$

where g(t) vanishes at least once in the interval [a,b]^[4][5] or where g(t) vanishes at a finite number of points in (a,b).^[6]

Limits of Integration[edit]

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 ${\displaystyle {\mathbb{�}}^{�}}$.^[3] Hence, the following two examples are Fredholm equations:^[1]

• Fredholm equation of the first type: ${\displaystyle �(�)={\int }_{�}^{�}�(�,�)�(�)��}$.
• Fredholm equation of the second type: ${\displaystyle �(�)=�(�)+�{\int }_{�}^{�}�(�,�)�(�)��.}$

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:

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

Hence, the above Fredholm equation of the second kind may be written compactly as:^[7]

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

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]

• Volterrra integral equation of the first kind: ${\displaystyle �(�)={\int }_{�}^{�}�(�,�)�(�)��}$
• Volterrra integral equation of the second kind: ${\displaystyle �(�)=�(�)+�{\int }_{�}^{�}�(�,�)�(�)��.}$

As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator ${\displaystyle \mathcal{�}:�(�)\to �(�)}$, as follows:^[3]

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

where ${\displaystyle �\in �=[{�}_0,�]}$ and K(t,s) is called the kernel and must be continuous on the interval ${\displaystyle �:=\{(�,�):0\le �\le �\le �\le \mathrm{\infty }\}}$.^[3] Hence, the Volterra integral equation of the first kind may be written as:^[3]

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

with ${\displaystyle �(0)=0}$. In addition, a linear Volterra integral equation of the second kind for an unknown function ${\displaystyle �(�)}$ and a given continuous function ${\displaystyle �(�)}$ on the interval ${\displaystyle �}$ where ${\displaystyle �\in �}$:

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

Volterra-Fredholm: In higher dimensions, integral equations such as Fredholm-Volterra integral equations (VFIE) exist.^[3] A VFIE has the form:

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

with ${\displaystyle �\in \mathrm{\Omega }}$ and ${\displaystyle \mathrm{\Omega }}$ being a closed bounded region in ${\displaystyle {\mathbb{�}}^{�}}$ with piecewise smooth boundary.^[3] The Fredholm-Volterrra Integral Operator ${\displaystyle \mathcal{�}:�(�\times \mathrm{\Omega })\to �(�\times \mathrm{\Omega })}$ is defined as:^[3]

$${\displaystyle (\mathcal{�}�)(�,�):={\int }_0^{�}{\int }_{\mathrm{\Omega }}�(�,�,�,�)�(�(�,�))����.}$$

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 ${\displaystyle [�,�]=�}$, but could also be defined over a curve or surface.^[7]