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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

Received  February 2021 Revised  May 2021 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021082
##### References:
  E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993. Google Scholar  G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992. Google Scholar  E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995. Google Scholar  G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001. Google Scholar  D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798.  Google Scholar  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.  Google Scholar  X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. Google Scholar  T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. Google Scholar  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.  Google Scholar  S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008. Google Scholar  T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. Google Scholar  R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014. Google Scholar  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.  Google Scholar  D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010. Google Scholar

show all references

##### References:
  E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993. Google Scholar  G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992. Google Scholar  E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995. Google Scholar  G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001. Google Scholar  D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798.  Google Scholar  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.  Google Scholar  X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. Google Scholar  T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. Google Scholar  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.  Google Scholar  S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008. Google Scholar  T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. Google Scholar  R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014. Google Scholar  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.  Google Scholar  D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010. Google Scholar 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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