Solving the Wave Equation on Discrete Time Scales

TL;DR

This paper develops a method to solve the 1D wave equation on various discrete and continuous time scales using Fourier transforms and contour integrals, extending classical solutions to more general settings.

Contribution

It introduces a novel approach for solving the wave equation on discrete time scales with Fourier analysis, broadening the scope beyond traditional continuous models.

Findings

Solution applicable to integer and broader discrete time scales

Utilizes Fourier transform and contour integrals for solving initial value problems

Extends classical wave equation solutions to discrete time scale settings

Abstract

This paper presents a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is set to the integers and a broader class of discrete time scales, while the time dimension is set to the positive real numbers.