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Fredholm Integral Equation of the First Kind


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

 f(x)=int_a^bK(x,t)phi(t)dt,
(1)

where K(x,t) is the kernel and phi(t) is an unknown function to be solved for (Arfken 1985, p. 865).

If the kernel is of the special form K(x-t) and the limits are infinite so that the equation becomes

 f(x)=int_(-infty)^inftyK(x-t)phi(t)dt,
(2)

then the solution (assuming the relevant transforms exist) is given by

 phi(x)=int_(-infty)^infty(F_x[f(x)](k))/(F_x[K(x)](k))e^(2piikx)dk,
(3)

where F_x is the Fourier transforms operator (Arfken 1985, pp. 875 and 877).


See also

Fredholm Integral Equation of the Second Kind, Integral Equation, 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, 1985.

Referenced on Wolfram|Alpha

Fredholm Integral Equation of the First Kind

Cite this as:

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

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