Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function

[EN] This paper deals with the approximate computation of the first probability density function of the solution stochastic process to second-order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses. Our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so-called Random Variable Transformation technique. The validity of the proposed method is shown through several illustrative examples.This work has been partially supported by the Ministerio de Econom ia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation. 331:33-45. https://doi.org/10.1016/j.amc.2018.02.051S334533

Solving random homogeneous linear second-order differential equations: a full probabilistic description

[EN] In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Second, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on random variable transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.This work has been partially supported by the Spanish M. C. Y. T. Grant MTM2013-41765-P.Casabán, M.; Cortés, J.; Romero, J.; Roselló, M. (2016). Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterranean Journal of Mathematics. 13(6):3817-3836. https://doi.org/10.1007/s00009-016-0716-6S38173836136Øksendal B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2007)Soong T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Neckel, T., Rupp, F.: Random Differential Equations in Scientific Computing. Versita, London (2013)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 1–18 (2014). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Chen-Charpentier, B.M.: A random differential transform method: theory and applications. Appl. Math. Lett. 25(10), 1490–1494 (2012). doi: 10.1016/j.aml.2011.12.033Licea, J.A., Villafuerte, L., Chen-Charpentier, B.M.: Analytic and numerical solutions of a Riccati differential equation with random coefficients. J. Comput. Appl. Math. 239, 208–219 (2013). doi: 10.1016/j.cam.2012.09.040Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random homogeneous linear second-order difference equations. Appl. Math. Lett. 34, 27–32 (2014). doi: 10.1016/j.aml.2014.03.010Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216 (5), 1524–1530 (2010). doi: 10.16/j.amc.2010.03.001El-Tawil, M., El-Tahan, W., Hussein, A.: Using FEM-RVT technique for solving a randomly excited ordinary differential equation with a random operator. Appl. Math. Comput. 187(2), 856–867 (2007). doi: 10.1016/j.amc.2006.08.164Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012). doi: 10.1016/j.amc.2011.12.088Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random SI-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 86–97 (2015). doi: 10.1016/j.cnsns.2014.12.016Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Determining the first probability density function of linear random initial value problems by the random variable transformation (RVT) technique: a comprehensive study. In: Abstract and Applied Analysis 2014-ID248512, pp. 1–25 (2014). doi: 10.1155/2013/248512Casabán, M.C., Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Villanueva, R.J.: A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 32, 199–210 (2016). doi: 10.1016/j.cnsns.2015.08.009El-Wakil, S.A., Sallah, M., El-Hanbaly, A.M.: Random variable transformation for generalized stochastic radiative transfer in finite participating slab media. Phys. A 435 66–79 (2015). doi: 10.1016/j.physa.2015.04.033Dorini, F.A., Cunha, M.C.C.: On the linear advection equation subject to random fields velocity. Math. Comput. Simul. 82, 679–690 (2011). doi: 10.16/j.matcom.2011.10.008Dhople, S.V., Domínguez-García, D.: A parametric uncertainty analysis method for Markov reliability and reward models. IEEE Trans. Reliab. 61(3), 634–648 (2012). doi: 10.1109/TR.2012.2208299Williams, M.M.R.: Polynomial chaos functions and stochastic differential equations. Ann. Nucl. Energy 33(9), 774–785 (2006). doi: 10.1016/j.anucene.2006.04.005Chen-Charpentier, B.M., Stanescu, D.: Epidemic models with random coefficients. Math. Comput. Model. 52(7/8), 1004–1010 (2009). doi: 10.1016/j.mcm.2010.01.014Papoulis A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1991