Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness

Abstract

[EN] This paper deals with the construction of numerical solutions of random hyperbolic models with a finite degree of randomness that make manageable the computation of its expectation and variance. The approach is based on the combination of the random Fourier transforms, the random Gaussian quadratures and the Monte Carlo method. The recovery of the solution of the original random partial differential problem throughout the inverse integral transform allows its numerical approximation using Gaussian quadratures involving the evaluation of the solution of the random ordinary differential problem at certain concrete values, which are approximated using Monte Carlo method. Numerical experiments illustrating the numerical convergence of the method are included.This work was partially supported by the Ministerio de Ciencia, Innovacion y Universidades Spanish grant MTM2017-89664-P.Casabán, M.; Company Rossi, R.; Jódar Sánchez, LA. (2019). Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness. Mathematics. 7(9):1-21. https://doi.org/10.3390/math7090853S12179Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving random mixed heat problems: A random integral transform approach. Journal of Computational and Applied Mathematics, 291, 5-19. doi:10.1016/j.cam.2014.09.021Casaban, M.-C., Cortes, J.-C., & Jodar, L. (2018). Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach. Mathematical Modelling and Analysis, 23(1), 79-100. doi:10.3846/mma.2018.006Saadatmandi, A., & Dehghan, M. (2010). Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numerical Methods for Partial Differential Equations, 26(1), 239-252. doi:10.1002/num.20442Weston, V. H., & He, S. (1993). Wave splitting of the telegraph equation in R 3 and its application to inverse scattering. Inverse Problems, 9(6), 789-812. doi:10.1088/0266-5611/9/6/013Jordan, P. M., & Puri, A. (1999). Digital signal propagation in dispersive media. Journal of Applied Physics, 85(3), 1273-1282. doi:10.1063/1.369258Banasiak, J., & Mika, J. R. (1998). Singularly perturbed telegraph equations with applications in the random walk theory. Journal of Applied Mathematics and Stochastic Analysis, 11(1), 9-28. doi:10.1155/s1048953398000021Kac, M. (1974). A stochastic model related to the telegrapher’s equation. Rocky Mountain Journal of Mathematics, 4(3), 497-510. doi:10.1216/rmj-1974-4-3-497Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher’s process. Statistics & Probability Letters, 55(1), 83-88. doi:10.1016/s0167-7152(01)00133-xCasabán, M.-C., Cortés, J.-C., & Jódar, L. (2015). A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259, 654-667. doi:10.1016/j.amc.2015.02.091Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics, 330, 937-954. doi:10.1016/j.cam.2016.11.049Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22), 9362-9377. doi:10.1016/j.apm.2016.06.01