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

TL;DR

The paper introduces a new method for solving first-kind integral equations with approximate inputs, ensuring solutions are close to true solutions without regularization, and demonstrates its effectiveness on various transform inversions.

Contribution

A novel approach that guarantees accurate solutions to integral equations with approximate data without regularization, using an ansatz for the solution's derivative.

Findings

Successfully applied to Lorentz, Stieltjes, and Laplace transforms

Achieves accurate solutions with small input deviations

No regularization needed for the method

Abstract

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.