Optimal Homotopy Asymptotic Method For Burgers Equation And Fokker-Planck Equation

Burgers equation and Fokker-Planck equation are common non-linear partial differential equations.They have important applications in physics and engineering.Homotopy analysis method is one of the common methods for solving approximate solution of non-linear problems.The characteristic of homotopy analysis method is to express the solution of non-linear problem as an infinite series form of a suitable basis function.Homotopy analysis method is to introduce an embedded variable that intervenes between 0 and 1 to get Zero order deformation equation and High order deformation equation by constituting homotopy function,constitute infinite number by changing the original non-linear equations into High order deformation equation,and take the sum of the former sub-problem’s solution to approach the access to exact solution.Compared with the traditional homotopy analysis method,the optimal homotopy asymptotic method adopted in this paper has the advantage that it does not need to determine the value of auxiliary function h in advance,but uses the least square method to determine the optimal value of auxiliary parameters,which is similar to the traditional homotopy method.The built-in convergence criterion of the analysis method is more flexible.The optimal homotopy asymptotic method is applied to solve the approximate series solution of the non-linear Burgers equation,linear Fokker-Planck equation and non-linear Fokker-Planck equation.The numerical examples are implemented to verify the effectiveness of the optimal homotopy asymptotic method.Moreover,it is more accurate than the approximate solution obtained from homotopy analysis method and homotopy perturbation method.