Method to solve integral equations of the first kind with an approximate input

Techniques are proposed for solving integral equations of the first kind with an input known not precisely. The requirement that the solution sought for includes a given number of maxima and minima is imposed. It is shown that when the deviation of the approximate input from the true one is sufficiently small and some additional conditions are fulfilled the method leads to an approximate solution that is necessarily close to the true solution. No regularization is required in the present approach. Requirements on features of the solution at integration limits are also imposed. The problem is treated with the help of an ansatz proposed for the derivative of the solution. The ansatz is the most general one compatible with the above mentioned requirements. The techniques are tested with exactly solvable examples. Inversions of the Lorentz, Stieltjes and Laplace integral transforms are performed, and very satisfactory results are obtained. The method is useful, in particular, for the calculation of quantum-mechanical reaction amplitudes and inclusive spectra of perturbation-induced reactions in the framework of the integral transform approach.

💡 Research Summary

The paper introduces a novel numerical technique for solving first‑kind integral equations when the input data are only approximately known. Traditional approaches to such equations rely on regularization methods (e.g., Tikhonov, truncated singular‑value decomposition) to suppress the inherent instability caused by small perturbations in the right‑hand side. Regularization, however, introduces an extra parameter that must be tuned and can obscure physically relevant features of the solution.

The key idea of the present work is to constrain the solution space by fixing the number of extrema (maxima and minima) that the true solution possesses. By assuming that the correct number of extrema, N, is known (or can be guessed from physical insight), the method searches only within the class of functions that have exactly N turning points. This restriction eliminates spurious narrow peaks or rapid oscillations that typically arise when an approximate input is inverted without additional information.

To implement the constraint, the authors work with the derivative of the unknown function, (f’(E)), and propose a highly flexible ansatz:

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