Discrete Bessel functions and discrete wave equation

TL;DR

This paper investigates discrete Bessel functions derived from discretized Bessel equations, analyzes their properties, and applies them to solve the discrete wave equation, revealing oscillatory solutions with decaying amplitude over time.

Contribution

It introduces and studies discrete Bessel functions with backward difference, deriving their solutions, properties, and applications to the discrete wave equation.

Findings

Discrete Bessel functions are solutions to discretized Bessel equations.

The fundamental solution of the discrete wave equation oscillates with exponentially decaying amplitude.

Explicit expressions for solutions of the discrete wave equation in terms of discrete Bessel functions.

Abstract

In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We focus on the discrete Bessel equations with the backward difference and derive their solutions. We then study the transformation properties of those functions, describe their asymptotic behaviour and compute Laplace transform. As an application, we study the discrete wave equation on the integers in timescale $T=\mathbb{Z}$ and express its fundamental and general solution in terms of the discrete $J$-Bessel function. Going further, we show that the first fundamental solution of this equation oscillates with the exponentially decaying amplitude as time tends to infinity.