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.
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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.
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.
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 Mi,j and vector vi 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.
See also
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
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
Categories: Integral equations
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