On semidiscrete models dominated by the heat, wave and Laplace equations

Carlos Lizama; Marina Murillo-Arcila

doi:10.3934/dcds.2024031

Article Contents

2024, Volume 44, Issue 8: 2368-2386. Doi: 10.3934/dcds.2024031

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On semidiscrete models dominated by the heat, wave and Laplace equations

Carlos Lizama^1, and
Marina Murillo-Arcila^{1, 2, ,}

1.
Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile
2.
Universidad de Cádiz, Facultad de Ciencias, Departamento de Matemáticas, E-11519, Puerto Real, Spain

^*Corresponding author: Marina Murillo-Arcila
^*Corresponding author: Marina Murillo-Arcila

Dedicated to the memory of my beloved sister Olga Lizama (1959-2024)

Received: November 2023

Revised: February 2024

Early access: March 2024

Published: August 2024

C. Lizama is partially supported by ANID Project FONDECYT 1220036 and DICYT, Universidad de Santiago de Chile, USACH. M. Murillo-Arcila is supported by MCIN/AEI/10.13039/501100011033, Projects PID2019-105011GBI00 and PID2022-139449NB-I00, by Generalitat Valenciana, Project PROMETEU/2021/070. M. Murillo-Arcila is also supported by the grant "Operator Theory: an interdisciplinary approach", reference ProyExcel_00780, a project financed in the 2021 call for Grants for Excellence Projects, under a competitive bidding regime, aimed at entities qualified as Agents of the Andalusian Knowledge System, in the scope of the Andalusian Research, Development and Innovation Plan (PAIDI 2020). Counseling of University, Research and Innovation of the Junta de Andalucía

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Abstract

We introduce a method to solve linear semidiscrete equations that provides for the first time explicit solutions for some well-known models such as the semidiscrete advection-diffusion equation and the semidiscrete Lighthill-Whitham-Richards equation, among others. We find conditions in the parameters of the model under which it can be dominated by the heat, wave or Laplace equations. We illustrate the fundamental solutions for all the cases based on the newly developed solution formulas. We also study spatial and time regularity in Lebesgue spaces for these models.

Keywords:

Discrete Laplacian,
Bessel functions,
semidiscrete heat,
wave and Laplace equations,
Lebesgue regularity,
semigroups and cosine functions of operators.

Mathematics Subject Classification: Primary: 39A06, 39A30; Secondary: 9A60, 47D06, 47D09.

Citation:

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Figure 1. The fundamental solution of (22) for $ \alpha = \beta = 1 $ and $ n = -2, ..., 2 $

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Figure 2. The fundamental solution of (42) for $ \alpha=\beta=1 $ and $ c=5/4 $ with $ n=-2,...,2 $

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Figure 3. The fundamental solution of (42) for $ \alpha=9/4 $, $ \beta=1/4 $ and $ c=1/4 $ with $ n=-2,...,2 $

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Figure 4. The fundamental solution of (56) for $ \alpha = \beta = 1/4 $ and $ c = 0 $ with $ n = 0, ..., 5 $

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