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The purpose of this thesis is to compare methods for solving the Fredholm integral equation of the first kind. The Fredholm equation has several practical applications including geology, superconductivity, and aerodynamics. Of specific interest is its application to determining radiation spectra using data from underground nuclear effects simulations. The two basic solution methods studied were the basis function and the iteration methods. The basis function method is a representation of the unfolded spectrum by a series of Planckian or cubic spline functions. The iteration method scales the unfolded spectrum so that its weighted integral over a given interval matches that of the actual spectrum. Both basis function methods produced excellent results when the actual spectrum was a sum of it's basis functions. The cubic spline method produced unfolded spectra which were good approximations for discontinuous actual spectra. However, there was a significant dropoff of the spectrum for the cubic spline for higher energies. The iteration method produced accurate approximations for actual spectra that were both basis function and discontinuous spectra. There were two problems with this method: the unfolded spectra were discontinuous at the discontinuities of the weighting function and noisy data sometimes produced large discontinuities in the unfolded spectra.