This paper introduces a new technique called Homotopy analysis Wiener Hermite expansion (HAM-WHE) which
considered as an extension to Wiener Hermite expansion linked with perturbation technique WHEP. The WHEP technique uses the
Wiener Hermite expansion and perturbation technique to solve a class of nonlinear partial differential equations with a perturbed
nonlinearity. The homotopy perturbation method (HPM) was used instead of the conventional perturbation methods which generalizes
the WHEP technique such that it can be applied to stochastic differential equations without the necessary of presence of the small
parameter. For more generalizing, the homotopy analysis method (HAM) is used instead of HPM; since HAM contains the control
parameter to guarantee the convergence of the solution and HPM is only a special case of HAM obtained at h¯ = −1 .The proposed
technique is applied on stochastic quadratic nonlinear diffusion problem to obtain some approximation orders of mean and variance
with making comparisons with HAM and homotopy-WHE to testify the method of analysis using symbolic computation software
Mathematica. The current work extends the use of WHEP for solving stochastic nonlinear differential equations. |
