Integral Equation

An equation involving a function  and integrals of that function to solved for . If the limits of the integral are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it is called a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be of the "first kind." If the function is both inside and outside, the equation is called of the "second kind." An example integral equation is given by

(1)

(Kress 1989, 1998), which has solution .

Let  be the function to be solved for,  a given known function, and  a known integral kernel. A Fredholm integral equation of the first kind is an integral equation of the form

(2)

A Fredholm integral equation of the second kind is an integral equation of the form

(3)

A Volterra integral equation of the first kind is an integral equation of the form

(4)

A Volterra integral equation of the second kind is an integral equation of the form

(5)

An integral equation is called homogeneous if .

Of course, not all integral equations can be written in one of these forms. An example that is close to (but not quite) a homogeneous Volterra integral equation of the second kind is given by the Dickman function

(6)

which fails to be Volterra because the integrand contains  instead of just .

Integral equations may be solved directly if they are separable. A integral kernel is said to separable if

(7)

This condition is satisfied by all polynomials.

Another general technique that may be used to solve an integral equation of the second kind (either Fredholm or Volterra) is an integral equation Neumann series (Arfken 1985, pp. 879-882).

REFERENCES:

Arfken, G. "Integral Equations." Ch. 16 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 865-924, 1985.

Corduneanu, C. Integral Equations and Applications. Cambridge, England: Cambridge University Press, 1991.

Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, 1962.

Kondo, J. Integral Equations. Oxford, England: Clarendon Press, 1992.

Kress, R. Linear Integral Equations. New York: Springer-Verlag, 1989.

Kress, R. Numerical Analysis. New York: Springer-Verlag, 1998.

Lovitt, W. V. Linear Integral Equations. New York: Dover, 1950.

Mikhlin, S. G. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd rev. ed. New York: Macmillan, 1964.

Mikhlin, S. G. Linear Integral Equations. New York: Gordon & Breach, 1961.

Pipkin, A. C. A Course on Integral Equations. New York: Springer-Verlag, 1991.

Polyanin, A. D. and Manzhirov, A. V. Handbook of Integral Equations. Boca Raton, FL: CRC Press, 1998.

Porter, D. and Stirling, D. S. G. Integral Equations: A Practical Treatment, from Spectral Theory to Applications. Cambridge, England: Cambridge University Press, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Integral Equations and Inverse Theory." Ch. 18 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 779-817, 1992.

Tricomi, F. G. Integral Equations. New York: Dover, 1957.

Weisstein, E. W. "Books about Integral Equations." http://www.ericweisstein.com/encyclopedias/books/IntegralEquations.html.

Whittaker, E. T. and Robinson, G. "The Numerical Solution of Integral Equations." §183 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 376-381, 1967.