Optimal Homotopy Analysis Method And Its Applications

Nonlinear equations exist many professional fields. How to solve them is an importantproblem in mathematics. Homotopy analysis method is a significant analytical method forsolving approximate series solutions of nonlinear equations. The method has been utilized tosolve many nonlinear di?erential and algebraic equations. The approximate analytical solutionby using the method includes a parameter which may control and adjust convergent range andrate of solution to a certain extent. How to select the parameter suitably is a basic problemin homotopy analysis method. In this paper, new methods for selecting convergent controlparameter in homotopy analysis method are proposed, and homotopy analysis method andNewton homotopy analysis method for solving general nonlinear algebraic are constructed.In Chapter 1, we concisely introduce derivation and current status of homotopy analysismethod, including developments on obtaining convergent control parameter and applicationson solving nonlinear equation by using homotopy analysis method.In Chapter 2, based on Marinca and one-step optimal homotopy analysis methods, multi-step and hybrid optimal homotopy analysis method are presented. Marinca and one-step op-timal method are special cases of the two new optimal method. We describe their homotopyanalysis process and computational results by utilizing the two new methods to solve a simplelinear di?erential equation. In order to observe computational e?ciency of the new methodsmore clearly, furthermore, we use them to solve a linear di?erential equation with variable coe?-cient and two nonlinear di?erential equations which are equivalent to two variational problems,respectively. Computational results show that two-step and hybrid optimal method can ob-tain high order approximate analytical solutions of these problems, and are much more e?cientthan Marinca and traditional methods. In addition, their total computational e?ects are almostequivalent to one of one-step optimal method.In Chapter 3, homotopy analysis method for solving general nonlinear algebraic equationis presented. The method includes many existing iterative approaches. Based on the newhomotopy analysis method, several iterative formulas, including sixth-order convergent methodand Newton homotopy analysis method, are educed. The convergent orders of these approachesare proved. Numerical results show that their computational e?ects are very good, and thatthe iterative approaches with convergent control parameter may increase the range of initialvalue and the e?ciency of convergence to some extent.In Chapter 4, we summarized the whole paper, and discussed some unsolved problems inhomotopy analysis method.