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I want to numerically solve Fredholm integral equations of the first kind, equations of the form

$$g(t)=\int_a^b K(t,s)f(s)\,\mathrm{d}s$$

where we know the functions $K(t,s)$ and $g(t)$ and seek to find $f(s)$. I'm aware of some papers presenting numerical methods (e.g. 1) but have not found any Mathematica implementations for what appears to be a reasonably standard problem.

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    $\begingroup$ The approaches described here: mathematica.stackexchange.com/questions/4677/… should help. $\endgroup$ – Mike Honeychurch Oct 9 '14 at 9:09
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    $\begingroup$ I defer to the mathematicians in the group, but the Volterra integral equation mentioned in the duplicate question is a Fredholm integral equation of the second kind, and of special convolution form. That equation has an unknown left-hand side which also appears under the integral. The Fredholm integral equation of the first kind in this question has a known left-hand side which does not appear under the integral, and is not explicitly of convolution form. Duplicate? $\endgroup$ – KennyColnago Oct 10 '14 at 17:18
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In addition to the reference provided by @MikeHoneychurch, your question about seeking $f(s)$ given the moments or projections or linear combinations $g(t)$, falls into the same category as this question about multi-peak fitting. There, the $f(s)$ are Gaussian-like shapes.

Are you working with discrete samplings of the data $g(t)$? or are you looking for theoretical, continuous forms of $g(t)$?

If discrete, then the integral equation may be transformed to a matrix system by partitioning the $s$-axis into discrete intervals. The function $f(s)$ may be represented on the partitioned $s$-axis as a comb of delta-functions, or narrow rectangular shapes, or other form. Finding the integrals with the known kernel functions $K(t,s)$ generates a matrix system $A\cdot x=b$. Mathematica has many built-in functions for solving discrete system, with and without constraints.

I have been working with such equations for many years, especially for systems where $x$ is non-negative. The book, Solving Least Squares Problems, and fortran code by Lawson and Hanson may be useful. Michael Woodhams implemented a key routine, NNLS, in Mathematica here. The paper by Istratov and Vyvenko (Exponential analysis in physical phenomena. Rev. Sci. Instrum. 70(2), 1233 (1999)) is a good review of techniques.

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