Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwell's equations.

Overview

The most basic type of integral equation is called a Fredholm equation of the first type,

f(x)=\int _{a}^{b}K(x,t)\,\varphi (t)\,dt.

The notation follows Arfken. Here $φ$ is an unknown function, $f$ is a known function, and $K$ is another known function of two variables, often called the kernel function. Note that the limits of integration are constant: this is what characterizes a Fredholm equation.

If the unknown function occurs both inside and outside of the integral, the equation is known as a Fredholm equation of the second type,

\varphi (x)=f(x)+\lambda \int _{a}^{b}K(x,t)\,\varphi (t)\,dt.

The parameter $λ$ is an unknown factor, which plays the same role as the eigenvalue in linear algebra.

If one limit of integration is a variable, the equation is called a Volterra equation. The following are called Volterra equations of the first and second types, respectively,

f(x)=\int _{a}^{x}K(x,t)\,\varphi (t)\,dt

\varphi (x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,\varphi (t)\,dt.

In all of the above, if the known function $f$ is identically zero, the equation is called a homogeneous integral equation. If $f$ is nonzero, it is called an inhomogeneous integral equation.

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 (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

\sum _{{j=1}}^{n}w_{j}K\left(s_{i},t_{j}\right)u(t_{j})=f(s_{i}),\qquad i=0,1,\cdots ,n.

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

u(t_{0}),u(t_{1}),\cdots ,u(t_{n}).

Classification

Integral equations are classified according to three different dichotomies, creating eight different kinds:

Limits of integration

both fixed: Fredholm equation
one variable: Volterra equation

Placement of unknown function

only inside integral: first kind
both inside and outside integral: second kind

Nature of known function $f$

identically zero: homogeneous
not identically zero: inhomogeneous

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.

Both Fredholm and Volterra equations are linear integral equations, due to the linear behaviour of $φ (x)$ under the integral. A nonlinear Volterra integral equation has the general form:

\varphi (x)=f(x)+\lambda \int _{a}^{x}K(x,t)\,F(x,t,\varphi (t))\,dt,

where $F$ is a known function.

Wiener–Hopf integral equations

Main article: Wiener–Hopf method

y(t)=\lambda x(t)+\int _{0}^{{\infty }}k(t-s)x(s)ds,\qquad 0\leq t<\infty .

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.

Power series solution for integral equations

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

g(s)=s\int _{{0}}^{{\infty }}dtK(st)f(t)

in the form of a power series

f(t)=\sum _{n=0}^{\infty }{\frac {a_{n}}{M(n+1)}}t^{n}

where

g(s)=\sum _{{n=0}}^{{\infty }}a_{{n}}s^{{-n}},\qquad M(n+1)=\int _{{0}}^{{\infty }}dtK(t)t^{{n}}

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

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

\sum _{j}M_{{i,j}}v_{j}=\lambda v_{i}

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

\int K(x,y)\varphi (y){\mathrm {d}}y=\lambda \varphi (x),

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.

Further information: Fredholm theory

References

Kendall E. Atkinson The Numerical Solution of integral Equations of the Second Kind. Cambridge Monographs on Applied and Computational Mathematics, 1997.
George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.
Harry Bateman (1910) History and Present State of the Theory of Integral Equations, Report of the British Association.
Andrei D. Polyanin and Alexander V. Manzhirov Handbook of Integral Equations. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4.
E. T. Whittaker and G. N. Watson. A Course of Modern Analysis Cambridge Mathematical Library.
M. Krasnov, A. Kiselev, G. Makarenko, Problems and Exercises in Integral Equations, Mir Publishers, Moscow, 1971
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Chapter 19. Integral Equations and Inverse Theory". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

External links

Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
Integral Equations: Index at EqWorld: The World of Mathematical Equations.
Hazewinkel, Michiel, ed. (2001), "Integral equation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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