Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods

Abstract

In this paper, the cubic and quintic diffusion equation under stochastic non homogeneity is solved using Wiener-Hermite expansion and perturbation (WHEP) technique, Homotopy perturbation method (HPM) and Pickard approximation technique. The analytic solution of the linear case is obtained using Eigenfunction expansion .The Picard approximation method is used to introduce the first and second order approximate solution for the non linear case. The WHEP technique is also used to obtain approximate solution under different orders and different corrections. The Homotopy perturbation method (HPM) is also used to obtain some approximation orders for mean and variance. Using mathematica-5, the methods of solution are illustrated through figures, comparisons among different methods and some parametric studies.

Solution of Nonlinear Stochastic Langevin’s Equation Using WHEP, Pickard and HPM Methods

“…The rate of convergence of HPM depends greatly on the initial approximation 0 υ which is considered as the main disadvantage of HPM [9][10][11]. The idea of the embedded parameter can be utilized to solve nonlinear problems by embedding this parameter to the problem and then forcing it to be unity in the obtained approximate solution if the convergence can be assured.…”

“…They have developed a theory of turbulence involving a truncated WHE of the velocity field. This technique was also used by M. El-Tawil and his co-workers [6][7][8][9][10].…”

“…A. Georges [4]. The Langevin's equation [5] with square nonlinear losses and stochastic non-homogeneity is solved by using different techniques, mainly the WHEP technique [6][7][8][9][10][11], the Picard approximation [8][9][10] and HPM [9][10][11].…”

Application of ANNs approach for wave-like and heat-like equations

“…During recent decades, various numerical techniques have been used for approximating different types of differential equations in linear and nonlinear cases. Among these, we can mention the homotopy perturbation method [2,4,5], variational iteration method [21,22,26], Adomian decomposition method [1,3,10], etc. An effective method in solving a wide variety of complicated mathematical problems is artificial neural networks (ANNs) approach.…”

Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

“…The application of the WHE [4][5][6][7][8][9][10] aims at finding a truncated series solution to the solution process of a stochastic differential equation. The truncated series is composed of two major parts: the first is the Gaussian part which consists of the first two terms, while the rest of the series constitutes the non-Gaussian part.…”