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Fredholm Integral Equation of the Second Kind


An integral equation of the form

 phi(x)=f(x)+lambdaint_(-infty)^inftyK(x,t)phi(t)dt
(1)
 phi(x)=1/(sqrt(2pi))int_(-infty)^infty(F(t)e^(-ixt)dt)/(1-sqrt(2pi)lambdaK(t)).
(2)

The solution to a general Fredholm integral equation of the second kind is called an integral equation Neumann series.

A Fredholm integral equation of the second kind with separable integral kernel may be solved as follows:

phi(x)=f(x)+int_a^bK(x,t)phi(t)dt
(3)
=f(x)+lambdasum_(j=1)^(n)M_j(x)int_a^bN_j(t)phi(t)dt
(4)
=f(x)+lambdasum_(j=1)^(n)c_jM_j(x),
(5)

where

 c_j=int_a^bN_j(t)phi(t)dt.
(6)

Now multiply both sides of (◇) by N_i(x) and integrate over dx.

 int_a^bphi(x)N_i(x)dx=int_a^bf(x)N_i(x)dx+lambdasum_(j=1)^nc_jint_a^bM_j(x)N_i(x)dx.
(7)

By (◇), the first term is just c_i. Now define

b_i=int_a^bN_i(x)f(x)dx
(8)
a_(ij)=int_a^bN_i(x)M_j(x)dx,
(9)

so (◇) becomes

 c_i=b_i+lambdasum_(j=1)^na_(ij)c_j.
(10)

Writing this in matrix form,

 C=B+lambdaAC,
(11)

so

 (I-lambdaA)C=B
(12)
 C=(I-lambdaA)^(-1)B.
(13)

See also

Fredholm Integral Equation of the First Kind, Integral Equation, Integral Equation Neumann Series, Volterra Integral Equation of the First Kind, Volterra Integral Equation of the Second Kind

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985.Baker, C. T. H. The Numerical Treatment of Integral Equations. Oxford, England: Clarendon Press, pp. 358-360, 1977.Pearson, C. E. Handbook of Applied Mathematics. New York: Van Nostrand, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Fredholm Equations of the Second Kind." §18.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 782-785, 1992.

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Fredholm Integral Equation of the Second Kind

Cite this as:

Weisstein, Eric W. "Fredholm Integral Equation of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FredholmIntegralEquationoftheSecondKind.html

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