The Optimal Homotopy Analysis Method For Nonlinear Problems

Nonlinear is everywhere. To solve nonlinear problems is an interesting work. The Homotopy Analysis Method is such a general analysis method for nonlinear problems. The basic idea of this popular method is to construct a family of equations(called the zeroth-order deformation equation) which connect the solution of nonlinear problem with an embedding parameter q∈[0,1]. When q changes from 0 to 1, the solution of deformation equation converges to the exact solution.Homotopy Analysis Method provides us with an effective way to control and adjust the convergence of the approximation series which contain an auxiliary param-eter h (or vector h) called convergence-control parameter(vector). It is possible to have the series solution converged in a wider region if we choose h (or vector h) properly.In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A general-ized homotopy equation is first expressed by means of an unknown embedding function in Taylor series, whose coefficients are then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach. It is shown that con-vergent series solution can be obtained by the optimal homotopy analysis method with in much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations efficiently.