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Integral Equation
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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).
SEE ALSO
Differential Equation, Fredholm Integral Equation of the First Kind,
Fredholm Integral Equation of the Second Kind, Integro-Differential
Equation, Volterra Integral Equation of the First Kind, Volterra Integral
Equation of the Second Kind
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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.
REFERENCED ON WOLFRAM|ALPHA
Integral Equation
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