Mean Square Convergence of the Numerical Solution of Random Differential Equations Article First Online: 12 August 2014 Received: 22 November 2013 Revised: 20 July 2014 Accepted: 22 July 2014 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 L _{2} -norm on Banach space and with probability one.

Mathematics Subject Classification Primary 60H10 Secondary 65L20 This research was in part supported by a grant from IPM (No. 91650054), and in part by the Research Council of Semnan University.

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