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Solving Nonlinear Diffusion Equations without Stochastic Homogeneity Using the Homotopy Perturbation Method

The study of random solutions of partial differential equations was initiated by Kampe de Feriet in 1955 [1]. In his valuable survey on the theory of random equations, Bharucha-Reid showed how a stochastic heat equation of Cauchy type can be solved using the stochastic integrals theory [2]. In 1973, Lo Dato V. [3] considered the stochastic velocity field and the Navier-Stokes equation and discussed the mathematical problems associated with it. Becus A. Georges [4] introduced a general solution for the heat conduction problem with a random source term and random initial and boundary conditions. Many authors investigated the stochastic diffusion equation under different views, see [5-11]. El-Tawil M. used the Wiener-Hermite expansion together with perturbation theory (WHEP technique) to solve a perturbed nonlinear stochastic diffusion equation [12]. The technique has been then developed to be applied on non-perturbed differential equations using the homotopy perturbation method and is called Homotopy-WHEP [13]. In this paper, the diffusion equation with nonlinear losses and stochastic no homogeneity are solved using Homotopy-WHEP. Some statistical moments are obtained, mainly the ensemble average and variance of the solution process. In section 2, the homotopy perturbation method is used alone in solving the nonlinear diffusion equation. The WHEP technique is used in section 3 for solving the nonlinear problem. In section 4, the Homotopy-WHEP technique is described and is used in solving the nonlinear diffusion case study with making comparisons.