Two Kinds Of Modified Homotopy Analysis Method

In the early1990s, Doc. Liao proposed the homotopy analysis method (HAM)based on homotopy to solve the nonlinear equation, a fundamental concept intopology and differential geometry. The method enjoys great freedom in choosinginitial approximate solution, auxiliary linear operators. At the same time, it has thefunction to regulate the region and the rate of convergence of the series. Because ofthe freedom and the advantages mentioned above, it has been applied to solvepractical problems more and more widely. But the existence of huge computationmakes the procedure complex in practical operation.To simplify the process of computation, this paper applies the iteration to theHAM, an iteration algorithm is presented to solve the nonlinear equations based onthe1st-order or2nd-order deformation equation of HAM. The new algorithm iscalled homotopy iteration method (HIM) which avoids the inherent hugecomputation of HAM. The solutions got by HIM are better than the ones got byHAM in some cases. Furthermore, the advantages of HAM will not be influenced atall.When solving nonlinear equations with initial conditions, the approximationobtained by HIM subjects to the distance factor. In order to solve this problem, thesubsection is applied to improve the HIM. The research range is divided into finitecells, and then the nonlinear equation is solved by HIM with approximate solution ofinterval left endpoints as the new initial condition on each interval. Experimentalresults show that the approximate solution is obviously improved.