Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation

[EN] In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X-0 and X-1. In a previous study (Calbo et al. in Comput Math Appl 61(9):2782-2792, 2011), a mean square convergent power series solution on (-1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X-0 and X-1, independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L-p solution (1 <= p <= infinity) to the random Legendre differential equation on the whole domain (-1, 1), as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of X-0 and X-1. Moreover, the growth condition on the moments of A is characterized by the boundedness of A, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation. Mediterranean Journal of Mathematics. 16(3):1-14. https://doi.org/10.1007/s00009-019-1338-6S114163Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Strand, J.L.: Random ordinary differential equations. J. Differ. Equ. 7(3), 538–553 (1970)Smith, R.C.: Uncertainty quantification. Theory, implementation, and application. 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Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme

[EN] We study the random heat partial differential equation on a bounded domain assuming that the diffusion coefficient and the boundary conditions are random variables, and the initial condition is a stochastic process. Under general conditions, this stochastic system possesses a unique solution stochastic process in the almost sure and mean square senses. To quantify the uncertainty for this solution process, the computation of the probability density function is a major goal. By using a random finite difference scheme, we approximate the stochastic solution at each point by a sequence of random variables, whose probability density functions are computable, i.e., we construct a sequence of approximating density functions. We include numerical experiments to illustrate the applicability of our method.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. The co-author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud, J.; Cortés, J.; Díaz, J.; Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics. 151:413-424. https://doi.org/10.1016/j.apnum.2020.01.012S413424151Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. 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