ANALYSIS AND APPLICATION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD AND THE USE OF DAFTARDAR- JAFFERY POLYNOMIALS

Abstract

In this thesis, we use the standard optimal homotopy asymptotic method for the analytical solution of higher order ordinary differential equations with two or multi- point boundary conditions, a nonlinear family of partial differential equations with an initial condition and Integro-differential equations. The results obtained are compared with the results obtained by the application of Adomain decomposition method, homotopy perturbation method, variational iteration method, differential transform method and homotopy analysis method etc. The optimal homotopy asymptotic method uses a flexible auxiliary function, which controls the convergence of the solution and allows adjustment in the convergence region where it is needed. Moreover, the procedure of this method is simple, clearly well-defined and the built- in recursive relations are explicitly worked out for easy use. Numerical results obtained by the optimal homotopy asymptotic method reveal high accuracy and excellent agreement with the exact solutions. Apart from the application of optimal homotopy asymtotic method, we develop a new scheme by using Daftardar-Jafari polynomials in the homotopy of optimal homotopy asymptotic method to solve nonlinear problems more effectively. This scheme is almost as simple as the optimal homotopy asymtotic method but its results are more accurate than the usual optimal homotopy asymptotic method. To show the effectiveness of this scheme, we solve different bench mark problems from literature and compare the results with those obtained by the standard optimal homotopy asymptotic method. To determine the convergence control parameters of the auxiliary function, two well known methods are followed: The Least squares and the Galerkin methods. We also explore and use a variety of the forms of auxiliary functions that results more accuracy and shows flexibility and reliability of the methods. For symbolic computation, we use Mathematica 7. The work presented in chapters 2, 3 and 4 of this thesis has been published in different reputed international journals and the work presented in chapters 3, 5, 6 and 7 is submitted for possible publication. The details of published/submitted work are included in the list of publications.