Physics – Mathematical Physics
Scientific paper
2011-10-16
Physics
Mathematical Physics
23 pages, 1 figure
Scientific paper
Techniques are given to solve integral equations of the first kind with an input known approximately. The requirement that the solution sought for includes a given number of maxima and minima is imposed. This removes the instability inherent to integral equations of the first kind. Requirements governing the behavior of the solution at integration limits are also imposed. The solution is found with the help of an ansatz proposed for its derivative. This is the most general anzatz compatible with the above mentioned requirements. Parameters entering it are fixed by fitting to an approximate input. A preliminary minimization with respect to parameters is performed employing discrete grids put one inside another. This provides good starting values for the conventional minimization. No regularization is required at performing the fit. The techniques are tested with an exactly solvable example. Inversions of the Lorentz, Stieltjes and Laplace integral transforms are performed, and very satisfactory results are obtained. The method is useful, in particular, to calculate quantum--mechanical reaction amplitudes and inclusive spectra of perturbation--induced reactions in the framework of the integral transform approach.
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