• Previous Article
    A drift homotopy implicit particle filter method for nonlinear filtering problems
  • DCDS-S Home
  • This Issue
  • Next Article
    Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species
doi: 10.3934/dcdss.2021082
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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:
[1]

E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007.  Google Scholar

[2]

P. AlmenarL. 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

[3]

H. T. BanksJ. L. DavisS. L. ErnstbergerS. HuE. ArtimovichA. 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

[4]

J. C. CortésP. 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

[5]

J. CalatayudJ. 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

[6]

J. C. CortésL. JódarL. 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

[7]

J. CalatayudJ. 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

[8]

S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993.  Google Scholar

[9]

G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992.  Google Scholar

[10] E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995.   Google Scholar
[11] G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001.   Google Scholar
[12]

D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798.  Google Scholar

[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.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. Google Scholar

[15]

T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. Google Scholar

[16]

F. RodríguezM. 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

[17]

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008.  Google Scholar

[18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973.   Google Scholar
[19]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014.  Google Scholar

[20]

L. VillafuerteC. A. BraumannJ. 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

[21] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010.   Google Scholar

show all references

References:
[1]

E. Allen, Modeling With Itô Stochastic Differential Equations, Springer Science & Business Media, Dordrecht, Netherlands, 2007.  Google Scholar

[2]

P. AlmenarL. 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

[3]

H. T. BanksJ. L. DavisS. L. ErnstbergerS. HuE. ArtimovichA. 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

[4]

J. C. CortésP. 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

[5]

J. CalatayudJ. 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

[6]

J. C. CortésL. JódarL. 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

[7]

J. CalatayudJ. 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

[8]

S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover, New York, 1993.  Google Scholar

[9]

G. B. Folland, Fourier Analysis and Its Applications, Brooks, Pacific Grove, CA, Wadsworth, 1992.  Google Scholar

[10] E. A. González-Velasco, Fourier Analysis and Boundary Value Problems, Academic Press, New York, 1995.   Google Scholar
[11] G. R. Grimmet and D. R. Stirzaker, Probability and Random Process, Clarendon Press, Oxford, 2001.   Google Scholar
[12]

D. Henderson and P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, Singapore, 2006. doi: 10.1142/9789812774798.  Google Scholar

[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.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. Google Scholar

[15]

T. Neckel and F. Rupp, Random Differential Equations in Scientific Computing, Walter de Gruyter, 2013. Google Scholar

[16]

F. RodríguezM. 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

[17]

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory, Universitext, Springer-Verlag Italia, Milan, 2008.  Google Scholar

[18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973.   Google Scholar
[19]

R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014.  Google Scholar

[20]

L. VillafuerteC. A. BraumannJ. 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

[21] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010.   Google Scholar
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.
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.
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.
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.
[1]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[2]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[3]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[4]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099

[5]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103

[6]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078

[7]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185

[8]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875

[9]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[10]

Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1543-1551. doi: 10.3934/dcdss.2020087

[11]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[12]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521

[13]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021040

[14]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159

[15]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[16]

Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038

[17]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[18]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051

[19]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial & Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483

[20]

Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial & Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (30)
  • HTML views (104)
  • Cited by (0)

[Back to Top]