Work

Transient Waves in Linear Dispersive Media with Dissipation: An Approach Based on the Steepest Descent Path

Year: 2025

Type: article

Abstract: In the study of linear dispersive media, it is of primary interest to gain knowledge of the impulse response of the material. The standard approach to compute the response involves a Laplace transform inversion, i.e., the solution of a Bromwich integral, which can be a notoriously troublesome problem. In this paper we propose a novel approach to the calculation of the impulse response, based on the well-assessed method of the steepest descent path, which results in the replacement of the Bromwich integral with a real line integral along the steepest descent path. In this exploratory investigation, the method is explained and applied to the case study of the Klein–Gordon equation with dissipation, for which analytical solutions of the Bromwich integral are available, so as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists of replacing a Laplace transform inverse with a potentially much less demanding real line integral, the method presented here could be of general interest in the study of linear dispersive waves in the presence of dissipation, as well as in other fields in which Laplace transform inversion comes into play.

Source: Mathematics

DOI: 10.3390/math13213418

Authors: Francesco Mainardi, Andrea Mentrelli, Juan Luis González-Santander

Institutions: University of Bologna, Istituto Nazionale di Fisica Nucleare, Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Universidad de Oviedo

Language: English

Topic: Differential Equations and Boundary Problems

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Gruppo Nazionale per la Fisica Matematica, Nuclear Physics