Application of Optimal Homotopy Asymptotic Method to Initial and Boundary Value Problems

Abstract

The development of nonlinear science has grown an ever-increasing interest among scientists and engineers for analytical asymptotic techniques for solving nonlinear problems. Finding solutions to linear problems by means of computer is easier nowadays; however, it is still difficult to solve nonlinear problems numerically or theoretically. The reason is the use of iterative techniques in the various discretization methods or numerical simulations to find numerical solutions to nonlinear problems. Almost all iterative methods are sensitive to initial solutions; hence, it is very difficult to obtain converging results in cases of strong nonlinearity. The objective of this dissertation is to use Optimal Homotopy Asymptotic Method (OHAM), a new semi-analytic approximating technique, for solving linear and nonlinear initial and boundary value problems. The semi analytic solutions of nonlinear fourth order, eighth order, special fourth order and special sixth order boundary-value problems are computed using OHAM. Successful application of OHAM for squeezing flow is a major task in this study. This dissertation also investigates the effectiveness of OHAM formulation for Partial Differential Equations (Wave Equation and Korteweg de Vries). OHAM is independent of the free parameter and there is no need of the initial guess as there is in Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM) and Homotopy Analysis Method (HAM). OHAM works very well with large domains and provides better accuracy at lower-order of approximations. Moreover, the convergence domain can be easily adjusted. The results are compared with other methods like HPM, VIM and HAM, which reveal that OHAM is effective, simpler, easier and explicit.