Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

[EN] In this paper we study random non-autonomous second order linear differential equations by taking advantage of the powerful theory of random difference equations. The coefficients are assumed to be stochastic processes, and the initial conditions are random variables both defined in a common underlying complete probability space. Under appropriate assumptions established on the data stochastic processes and on the random initial conditions, and using key results on difference equations, we prove the existence of an analytic stochastic process solution in the random mean square sense. Truncating the random series that defines the solution process, we are able to approximate the main statistical properties of the solution, such as the expectation and the variance. We also obtain error a priori bounds to construct reliable approximations of both statistical moments. We include a set of numerical examples to illustrate the main theoretical results established throughout the paper. We finish with an example where our findings are combined with Monte Carlo simulations to model uncertainty using real data.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.; Villafuerte, L. (2018). Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Advances in Difference Equations. (3):1-29. https://doi.org/10.1186/s13662-018-1848-8S1293Apostol, T.M.: Mathematical Analysis, 2nd edn. Pearson, New York (1976)Boyce, W.E.: Probabilistic Methods in Applied Mathematics I. Academic Press, New York (1968)Calbo, G., Cortés, J.C., Jódar, L.: Random Hermite differential equations: mean square power series solutions and statistical properties. Appl. Math. Comput. 218(7), 3654–3666 (2011)Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Solving the random Legendre differential equation: mean square power series solution and its statistical functions. Comput. Math. Appl. 61(9), 2782–2792 (2011)Casabán, M.C., Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Villanueva, R.J.: Computing probabilistic solutions of the Bernoulli random differential equation. J. Comput. Appl. Math. 309, 396–407 (2017)Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterr. J. Math. 13(6), 3817–3836 (2016)Cortés, J.C., Jódar, L., Camacho, J., Villafuerte, L.: Random Airy type differential equations: mean square exact and numerical solutions. Comput. Math. Appl. 60(5), 1237–1244 (2010)Cortés, J.C., Jódar, L., Company, R., Villafuerte, L.: Laguerre random polynomials: definition, differential and statistical properties. Util. Math. 98, 283–295 (2015)Cortés, J.C., Jódar, L., Villafuerte, L.: Random linear-quadratic mathematical models: computing explicit solutions and applications. Math. Comput. Simul. 79(7), 2076–2090 (2009)Cortés, J.C., Jódar, L., Villafuerte, L.: Mean square solution of Bessel differential equation with uncertainties. J. Comput. Appl. Math. 309(1), 383–395 (2017)Cortés, J.C., Sevilla-Peris, P., Jódar, L.: Analytic-numerical approximating processes of diffusion equation with data uncertainty. Comput. Math. Appl. 49(7–8), 1255–1266 (2005)Díaz-Infante, S., Jerez, S.: Convergence and asymptotic stability of the explicit Steklov method for stochastic differential equations. J. Comput. Appl. Math. 291(1), 36–47 (2016)Dorini, F., Cunha, M.: Statistical moments of the random linear transport equation. J. Comput. Phys. 227(19), 8541–8550 (2008)Dorini, F.A., Cecconello, M.S., Dorini, M.B.: On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Commun. Nonlinear Sci. Numer. Simul. 33, 160–173 (2016)Golmankhaneh, A.K., Porghoveh, N.A., Baleanu, D.: Mean square solutions of second-order random differential equations by using homotopy analysis method. Rom. Rep. Phys. 65(2), 350–362 (2013)Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Clarendon Press, Oxford (2000)Henderson, D., Plaschko, P.: Stochastic Differential Equations in Science and Engineering. Cambridge Texts in Applied Mathematics. World Scientific, Singapore (2006)Hussein, A., Selim, M.M.: A developed solution of the stochastic Milne problem using probabilistic transformations. Appl. Math. Comput. 216(10), 2910–2919 (2009)Hussein, A., Selim, M.M.: Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique. Appl. Math. Comput. 213(1), 250–261 (2009)Hussein, 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)Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge–Kutta methods. Math. Comput. Model. 53(9–10), 1910–1920 (2011)Khodabin, M., Rostami, M.: Mean square numerical solution of stochastic differential equations by fourth order Runge–Kutta method and its application in the electric circuits with noise. Adv. Differ. Equ. 2015, 62 (2015)Khudair, A.K., Ameen, A.A., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci. 51(5), 2521–2535 (2011)Khudair, A.K., Haddad, S.A.M., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using the differential transformation method. Open J. Appl. Sci. 6, 287–297 (2016)Lesaffre, E., Lawson, A.B.: Bayesian Biostatistics. Statistics in Practice. Wiley, New York (2012)Li, X., Fu, X.: Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks. J. Comput. Appl. Math. 234(2), 407–417 (2010)Licea, J.A., Villafuerte, L., Chen-Charpentier, B.M.: Analytic and numerical solutions of a Riccati differential equation with random coefficients. J. Comput. Appl. Math. 309(1), 208–219 (2013)Liu, S., Debbouche, A., Wang, J.: On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017)Loève, M.: Probability Theory. Vol. I. Springer, Mineola (1977)Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Dover, New York (2014)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123–1140 (2015)Rencher, A.C., Schaalje, G.B.: Linear Models in Statistics, 2nd edn. Wiley, New York (2008)Santos, 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)Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Cambridge Texts in Applied Mathematics. Wiley, New York (2003)Smith, R.C.: Uncertainty Quantification: Theory, Implementation, and Applications. SIAM, Philadelphia (2014)Soheili, A.R., Toutounian, F., Soleymani, F.: A fast convergent numerical method for matrix sign function with application in SDEs (Stochastic Differential Equations). J. Comput. Appl. Math. 282, 167–178 (2015)Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010)Xiu, D.: Numerical Methods for Stochastic Computations. A Spectral Method Approach. Cambridge Texts in Applied Mathematics. Princeton University Press, New York (2010