Variational Iteration and Linearized Liapunov Methods for Seeking the Analytic Solutions of Nonlinear Boundary Value Problems

The boundary shape function method (BSFM) and the variational iteration method (VIM) are merged together to seek the analytic solutions of nonlinear boundary value problems. The boundary shape function method transforms the boundary value problem to an initial value problem (IVP) for a new variable. Then, a modified variational iteration method (MVIM) is created by applying the VIM to the resultant IVP, which can achieve a good approximate solution to automatically satisfy the prescribed mixed-boundary conditions. By using the Picard iteration method, the existence of a solution is proven with the assumption of the Lipschitz condition. The MVIM is equivalent to the Picard iteration method by a back substitution. Either by solving the nonlinear equations or by minimizing the error of the solution or the governing equation, we can determine the unknown values of the parameters in the MVIM. A nonlocal BSFM is developed, which then uses the MVIM to find the analytic solution of a nonlocal nonlinear boundary value problem. In the second part of this paper, a new splitting–linearizing method is developed to expand the analytic solution in powers of a dummy parameter. After adopting the Liapunov method, linearized differential equations are solved sequentially to derive an analytic solution. Accurate analytical solutions are attainable through a few computations, and some examples involving two boundary layer problems confirm the efficiency of the proposed methods.