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Discrete Bessel functions and discrete wave equation

Amar Bašić, Lejla Smajlović, Zenan Šabanac

TL;DR

The paper develops backward-difference discretizations of the Bessel equation on the integers, introducing backward-discrete $J$- and $I$-Bessel functions via Gauss hypergeometric representations and establishing their transformation, asymptotic, and Laplace-transform properties. It then applies these results to the backward discrete wave equation on $\mathbb{Z}$, deriving explicit fundamental and general solutions in terms of $\overline{J}_{n}^{2c}$, and analyzes their long-time oscillatory behavior with decaying amplitude. Through generating-function and asymptotic analyses, the work connects backward-discrete Bessel functions to classical Bessel transforms and Legendre functions, providing a coherent framework for discrete-time wave dynamics on $\mathbb{Z}$. The findings show that backward-time discretization leads to exponentially decaying amplitudes while forward-time discretization produces growth, offering insights for time-scale discretizations and numerical schemes in discrete wave propagation. Overall, the results advance the theory of discrete Bessel analogues and their role in discrete wave phenomena on integers, with potential applications in numerical analysis and time-scale calculus.

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.

Discrete Bessel functions and discrete wave equation

TL;DR

The paper develops backward-difference discretizations of the Bessel equation on the integers, introducing backward-discrete - and -Bessel functions via Gauss hypergeometric representations and establishing their transformation, asymptotic, and Laplace-transform properties. It then applies these results to the backward discrete wave equation on , deriving explicit fundamental and general solutions in terms of , and analyzes their long-time oscillatory behavior with decaying amplitude. Through generating-function and asymptotic analyses, the work connects backward-discrete Bessel functions to classical Bessel transforms and Legendre functions, providing a coherent framework for discrete-time wave dynamics on . The findings show that backward-time discretization leads to exponentially decaying amplitudes while forward-time discretization produces growth, offering insights for time-scale discretizations and numerical schemes in discrete wave propagation. Overall, the results advance the theory of discrete Bessel analogues and their role in discrete wave phenomena on integers, with potential applications in numerical analysis and time-scale calculus.

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 and express its fundamental and general solution in terms of the discrete -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.
Paper Structure (17 sections, 13 theorems, 84 equations)

This paper contains 17 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction and statement of results
  2. Discretizations of Bessel functions
  3. Discrete time wave equation on integers
  4. Organization of the paper
  5. Preliminaries
  6. Hypergeometric function
  7. Legendre function of the first kind
  8. Laplace transform for the forward and the backward difference
  9. Properties of backward discrete Bessel Functions
  10. Proofs of main results
  11. Proof of Theorem \ref{['thm: difference eq.']}
  12. Proof of Theorem \ref{['th_asymp']}
  13. Proof of Theorem \ref{['thm: Laplace tr']}
  14. Backward discrete wave equation
  15. Fundamental solutions to the backward discrete wave equation
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $c\in \mathbb{C}\setminus\{i\alpha:\, \alpha\in \mathbb{R},\, |\alpha|\geq 1\}$. Then, the function is the solution to the backward difference equation backBesselEq1, with the plus sign. For $c\in \mathbb{C}\setminus\{\alpha:\, \alpha\in \mathbb{R},\, |\alpha|\geq 1\}$, the function is the solution to the backward difference equation backBesselEq1, with the minus sign.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 10 more