Mediterranean Journal of Mathematics

, Volume 12, Issue 3, pp 1123–1140

Mean Square Convergence of the Numerical Solution of Random Differential Equations

Article

DOI: 10.1007/s00009-014-0452-8

Cite this article as:
Nouri, K. & Ranjbar, H. Mediterr. J. Math. (2015) 12: 1123. doi:10.1007/s00009-014-0452-8

Abstract

This paper is devoted to the construction of an approximate solution for the random differential equation with an initial condition and defined on a partition of the time-interval. We employ a random mean value theorem to achieve our goals in this work. The implicit Runge–Kutta method is presented and the conditions for the mean square convergence are established. Finally, illustrative examples are included in which the main statistical properties such as the mean and the variance of the random approximate solution process are given. The closeness of the original and approximate solutions is measured in the sense of the L2-norm on Banach space and with probability one.

Mathematics Subject Classification

Primary 60H10Secondary 65L20

Keywords

Random differential equationmean square solutionrandom implicit Runge–Kutta methodsrandom mean value theorem

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics, Statistics and Computer ScienceSemnan UniversitySemnanIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran