On the random wave equation within the mean square context

Julia Calatayud; Juan Carlos Cortés; Marc Jornet

doi:10.3934/dcdss.2021082

Article Contents

2022, Volume 15, Issue 2: 409-425. Doi: 10.3934/dcdss.2021082

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On the random wave equation within the mean square context

1.
Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain
2.
Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

^* Corresponding author: Juan Carlos Cortés
^* Corresponding author: Juan Carlos Cortés

Received: February 2021

Revised: May 2021

Early access: July 2021

Published: February 2022

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Abstract

This paper deals with the random wave equation on a bounded domain with Dirichlet boundary conditions. Randomness arises from the velocity wave, which is a positive random variable, and the two initial conditions, which are regular stochastic processes. The aleatory nature of the inputs is mainly justified from data errors when modeling the motion of a vibrating string. Uncertainty is propagated from these inputs to the output, so that the solution becomes a smooth random field. We focus on the mean square contextualization of the problem. Existence and uniqueness of the exact series solution, based upon the classical method of separation of variables, are rigorously established. Exact series for the mean and the variance of the solution process are obtained, which converge at polynomial rate. Some numerical examples illustrate these facts.

Keywords:

Random wave partial differential equation,
mean square calculus,
exact series solution,
separation of variables,
mean and variance.

Mathematics Subject Classification: Primary: 35C05, 35C10, 35R60.

Citation:

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Figure 1. Expectation and variance of the solution $ u(x,t) $ to (1), for different space-time points and orders of truncation $ N $ of the series (2). This figure corresponds to Example 1.

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Figure 2. Rate of convergence of $ \mathbb{E}[u_N(0.5,2)] $ and $ \mathbb{V}[u_N(0.5,2)] $ with $ N $, where $ u_N(x,t) $ is the truncation (11) of $ u(x,t) $ (2). This figure corresponds to Example 1.

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Figure 3. Expectation and variance of the solution $ u(x,t) $ to (1), for different space-time points and orders of truncation $ N $ of the series (2). This figure corresponds to Example 2.

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Figure 4. Rate of convergence of $ \mathbb{E}[u_N(0.5,2)] $ and $ \mathbb{V}[u_N(0.5,2)] $ with $ N $, where $ u_N(x,t) $ is the truncation (11) of $ u(x,t) $ (2). This figure corresponds to Example 2.

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