.

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, Maxwell's equations.

Overview

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

\( f(x) = \int \limits_a^b K(x,t)\,\varphi(t)\,dt. \)

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, it is known as a Fredholm equation of the second type:

\( \varphi(x) = f(x)+ \lambda \int \limits_a^b K(x,t)\,\varphi(t)\,dt. \)

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

If one limit of integration is variable, it is called a Volterra equation. Thus Volterra equations of the first and second types, respectively, would appear as:

\( f(x) = \int \limits_a^x K(x,t)\,\varphi(t)\,dt \)
\( \varphi(x) = f(x) + \lambda \int \limits_a^x K(x,t)\,\varphi(t)\,dt. \)

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

\( \varphi(x) = f(x) + \lambda \int \limits_a^x K(x,t)\,F(x, t, \varphi(t))\,dt. , \)

where F is a known function.
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 \mathbf{M} is a matrix,\( \mathbf{v} \) is one of its eigenvectors, and \lambda is the associated eigenvalue.

Taking the continuum limit, by replacing the discrete indices i and j with continuous variables x and y, gives

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

where the sum over j has been replaced by an integral over y and the \( matrix M_{i,j} \) and vector \( v_i \) have been replaced by the 'kernel' K(x,y) and the eigenfunction \( \varphi(y) \). (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.
References

George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.
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.

Integral Equations and Operator Theory

Integral equations with applications, Introduction to , Abdul J. Jerri