Fredholm Integral Equation - Equation of The First Kind

Equation of The First Kind

Integral equations, most generally, are common and take many specific forms (Fourier, Laplace, Hankel, etc.). They each differ in their kernels (defined below). What is distinctive about Fredholm integral equations is that they are integral equations in which the integration limits are constants (they do not include the variable). This is contrast to Volterra integral equations.

An inhomogeneous Fredholm equation of the first kind is written as:

and the problem is, given the continuous kernel function K(t,s), and the function g(t), to find the function f(s).

If the kernel is a function only of the difference of its arguments, namely, and the limits of integration are, then the right hand side of the equation can be rewritten as a convolution of the functions K and f and therefore the solution will be given by

f(t) = \mathcal{F}_\omega^{-1}\left[ {\mathcal{F}_t(\omega)\over \mathcal{F}_t(\omega)} \right]=\int_{-\infty}^\infty {\mathcal{F}_t(\omega)\over \mathcal{F}_t(\omega)}e^{2\pi i \omega t} \mathrm{d}\omega

where and are the direct and inverse Fourier transforms respectively.

Read more about this topic:  Fredholm Integral Equation

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