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A practical comparison between the spectral techniques in solving the SDEs

Mohamed El-Beltagy (Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Cairo, Egypt)

Engineering Computations

ISSN: 0264-4401

Article publication date: 13 August 2019

Issue publication date: 12 September 2019




The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes.


The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques.


The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems.

Research limitations/implications

Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions.


This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.



El-Beltagy, M. (2019), "A practical comparison between the spectral techniques in solving the SDEs", Engineering Computations, Vol. 36 No. 7, pp. 2369-2402.



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