Solving random homogeneous linear second-order differential equations: a full probabilistic description. (English) Zbl 1353.60051
Summary: 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 37H10 | Generation, random and stochastic difference and differential equations |
Keywords:
random differential equations; random variable transformation; probability density functionsReferences:
| [1] | Øksendal B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2007) · Zbl 1025.60026 |
| [2] | Soong T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973) · Zbl 0348.60081 |
| [3] | Neckel, T., Rupp, F.: Random Differential Equations in Scientific Computing. Versita, London (2013) · Zbl 1323.60002 |
| [4] | Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 1-18 (2014). doi:10.1007/s00009-014-0452-8 |
| [5] | Villafuerte, 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.2011.12.033 · Zbl 1251.65005 |
| [6] | 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. 239, 208-219 (2013). doi:10.1016/j.cam.2012.09.040 · Zbl 1255.65023 |
| [7] | Casabán, 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.2014.03.010 · Zbl 1376.37098 |
| [8] | 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). doi:10.16/j.amc.2010.03.001 · Zbl 1201.65014 |
| [9] | El-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.2006.08.164 · Zbl 1123.65300 |
| [10] | 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). doi:10.1016/j.amc.2011.12.088 · Zbl 1246.65014 |
| [11] | Casabán, 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.2014.12.016 · Zbl 1440.92061 |
| [12] | Casabán, 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:10.1155/2013/248512 · Zbl 1470.60195 |
| [13] | Casabán, 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.2015.08.009 · Zbl 1510.92202 |
| [14] | El-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.2015.04.033 · Zbl 1459.78009 |
| [15] | Dorini, 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.2011.10.008 · Zbl 1241.35224 |
| [16] | Dhople, 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.2012.2208299 |
| [17] | Williams, M.M.R.: Polynomial chaos functions and stochastic differential equations. Ann. Nucl. Energy 33(9), 774-785 (2006). doi:10.1016/j.anucene.2006.04.005 |
| [18] | Chen-Charpentier, B.M., Stanescu, D.: Epidemic models with random coefficients. Math. Comput. Model. 52(7/8), 1004-1010 (2009). doi:10.1016/j.mcm.2010.01.014 · Zbl 1205.60127 |
| [19] | Papoulis A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1991) · Zbl 0191.46704 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
