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February  2022, 15(2): 409-425. doi: 10.3934/dcdss.2021082

## 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

Received  February 2021 Revised  May 2021 Published  February 2022 Early access  July 2021

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.

Citation: Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 409-425. doi: 10.3934/dcdss.2021082
##### References:
 [1] E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007. [2] P. Almenar, L. Jódar and J. A. Martín, Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds, Mathematical and Computer Modelling, 25 (1997), 31-44.  doi: 10.1016/S0895-7177(97)00082-4. [3] H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, A. K. Dhar and C. L. Browdy, A comparison of probabilistic and stochastic formulations in modelling growth uncertainty and variability, Journal of Biological Dynamics, 3 (2009), 130-148.  doi: 10.1080/17513750802304877. [4] J. C. Cortés, P. Sevilla-Peris and L. Jódar, Analytic-numerical approximating processes of diffusion equation with data uncertainty, Computers & Mathematics with Applications, 49 (2005), 1255-1266.  doi: 10.1016/j.camwa.2004.05.015. [5] J. Calatayud, J. C. Cortés and M. Jornet, Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function, Mathematical Methods in the Applied Sciences, 42 (2019), 5649-5667.  doi: 10.1002/mma.5333. [6] J. C. Cortés, L. Jódar, L. Villafuerte and F. J. Camacho, Random Airy type differential equations: Mean square exact and numerical solutions, Computers and Mathematics with Applications, 60 (2010), 1237-1244.  doi: 10.1016/j.camwa.2010.05.046. [7] J. Calatayud, J. C. Cortés and M. Jornet, Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: A comparative case study with random Fröbenius method and Monte Carlo simulation, Open Mathematics, 16 (2018), 1651-1666.  doi: 10.1515/math-2018-0134. [8] S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993. [9] G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992. [10] E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995. [11] G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001. [12] D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798. [13] L. Jódar and P. Almenar, Accurate continuous numerical solutions of time dependent mixed partial differential problems, Computers & Mathematics with Applications, 32 (1996), 5-19.  doi: 10.1016/0898-1221(96)00099-5. [14] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. [15] T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. [16] F. Rodríguez, M. Roales and J. A. Martín, Exact solutions and numerical approximations of mixed problems for the wave equation with delay, Applied Mathematics and Computation, 219 (2012), 3178-3186.  doi: 10.1016/j.amc.2012.09.050. [17] S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008. [18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. [19] R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014. [20] L. Villafuerte, C. A. Braumann, J. C. Cortés and L. Jódar, Random differential operational calculus: Theory and applications, Comput. Math. Appl., 59 (2010), 115-125.  doi: 10.1016/j.camwa.2009.08.061. [21] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010.

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##### References:
 [1] E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007. [2] P. Almenar, L. Jódar and J. A. Martín, Mixed problems for the time-dependent telegraph equation: Continuous numerical solutions with a priori error bounds, Mathematical and Computer Modelling, 25 (1997), 31-44.  doi: 10.1016/S0895-7177(97)00082-4. [3] H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, A. K. Dhar and C. L. Browdy, A comparison of probabilistic and stochastic formulations in modelling growth uncertainty and variability, Journal of Biological Dynamics, 3 (2009), 130-148.  doi: 10.1080/17513750802304877. [4] J. C. Cortés, P. Sevilla-Peris and L. Jódar, Analytic-numerical approximating processes of diffusion equation with data uncertainty, Computers & Mathematics with Applications, 49 (2005), 1255-1266.  doi: 10.1016/j.camwa.2004.05.015. [5] J. Calatayud, J. C. Cortés and M. Jornet, Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function, Mathematical Methods in the Applied Sciences, 42 (2019), 5649-5667.  doi: 10.1002/mma.5333. [6] J. C. Cortés, L. Jódar, L. Villafuerte and F. J. Camacho, Random Airy type differential equations: Mean square exact and numerical solutions, Computers and Mathematics with Applications, 60 (2010), 1237-1244.  doi: 10.1016/j.camwa.2010.05.046. [7] J. Calatayud, J. C. Cortés and M. Jornet, Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: A comparative case study with random Fröbenius method and Monte Carlo simulation, Open Mathematics, 16 (2018), 1651-1666.  doi: 10.1515/math-2018-0134. [8] S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993. [9] G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992. [10] E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995. [11] G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001. [12] D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798. [13] L. Jódar and P. Almenar, Accurate continuous numerical solutions of time dependent mixed partial differential problems, Computers & Mathematics with Applications, 32 (1996), 5-19.  doi: 10.1016/0898-1221(96)00099-5. [14] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. [15] T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. [16] F. Rodríguez, M. Roales and J. A. Martín, Exact solutions and numerical approximations of mixed problems for the wave equation with delay, Applied Mathematics and Computation, 219 (2012), 3178-3186.  doi: 10.1016/j.amc.2012.09.050. [17] S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008. [18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. [19] R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014. [20] L. Villafuerte, C. A. Braumann, J. C. Cortés and L. Jódar, Random differential operational calculus: Theory and applications, Comput. Math. Appl., 59 (2010), 115-125.  doi: 10.1016/j.camwa.2009.08.061. [21] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010.
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.
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.
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.
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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