A Unified Numerical Framework for the Inversion of Integral Transforms: Laplace and Fourier

Abstract

The analytical inversion of Laplace and Fourier transforms often presents significant challenges for functions commonly encountered in engineering and scientific fields. Although numerous numerical inversion methods are available, these techniques tend to be highly specialized, algorithmically complex, and sensitive to parameter variations. This paper introduces an innovative, unified methodological framework for the numerical computation of these inverse transforms. By leveraging the intrinsic relationship between the two transforms through analytic continuation and implementing a robust regularization strategy within a Fourier inversion framework, we derive a generalized inversion formula. This method effectively converts the problem of inverting a Laplace transform into a specially designed Fourier inversion, which is then solved using a computationally efficient and stable algorithm based on the Fast Fourier Transform (FFT) with a smoothing kernel. We validate the effectiveness and precision of this universal method through several standard examples, comparing its performance with traditional techniques such as the Bromwich integral and Stehfest's algorithm for Laplace inversion. The proposed approach offers a streamlined, powerful, and versatile computational tool for researchers and practitioners.