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
Cite this as:
Weisstein, Eric W. "Integral Equation."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/IntegralEquation.html

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