The Application of Homotopy Analysis Method in Solving Stochastic Nonlinear Partial Differential Equations

Abstract

The homotopy analysis method (HAM) is one of the important analytical methods, which introduces a series solution for nonlinear problems. HAM contains the auxiliary parameter ħ which gives a way to adjust and control the convergence region of the series solution.

In this work, a new technique for using (HAM) together with Wiener- Hermite expansion (HAM-WHE) is proposed as an efficient method for solving stochastic nonlinear differential equations. Piecewise homotopy analysis method (P-HAM) is also introduced to increase the convergence region.

Burgers’ equation and diffusion equation are solved using HAM using the parameter ħ to guarantee the convergence of the series solutions of nonlinear differential equations. Langevin’s equation is solved using (P-HAM). The proposed technique (HAM-WHE) is used to solve stochastic quadratic nonlinear diffusion equation.