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Journal of Applied Mathematics
Volume 2013, Article ID 950469, 9 pages
Research Article

A Proposed Stochastic Finite Difference Approach Based on Homogenous Chaos Expansion

Engineering Mathematics and Physics Department, Faculty of Engineering, Fayoum University, Fayoum 63514, Egypt

Received 19 March 2013; Revised 24 June 2013; Accepted 26 June 2013

Academic Editor: Livija Cveticanin

Copyright © 2013 O. H. Galal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper proposes a stochastic finite difference approach, based on homogenous chaos expansion (SFDHC). The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using Karhunen-Loève expansion, while the response function is approximated using homogenous chaos expansion. Galerkin projection is used in converting the original stochastic partial differential equation (PDE) into a set of coupled deterministic partial differential equations and then solved using finite difference method. Two well-known equations were used for efficiency validation of the method proposed. First one being the linear diffusion equation with stochastic parameter and the second is the nonlinear Burger's equation with stochastic parameter and stochastic initial and boundary conditions. In both of these examples, the probability distribution function of the response manifested close conformity to the results obtained from Monte Carlo simulation with optimized computational cost.