Integral equation

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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.

Contents 1 Overview 2 Classification 3 Integral equations as a generalization of eigenvalue equations 4 See also 5 References 6 External links

[edit] Overview

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

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:

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:

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.

[edit] 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:

,

where F is a known function.

[edit] 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

,

where is a matrix, is one of its eigenvectors, and λ is the associated eigenvalue.

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

,

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 . (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.

[edit] See also

• Hilbert-Schmidt operator

[edit] 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

[edit] External links

• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Integral Equations: Index at EqWorld: The World of Mathematical Equations.
• Integral equations at exampleproblems.com