OPTIMAL HOMOTOPY ASYMPTOTIC METHOD TO SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

Abstract

Nonlinear Differential equations are of major importance in different fields of science and engineering. For complicated nonlinear problems exact solutions are not available and alternate way is to use numerical methods, Iterative methods or analytical techniques of perturbation. Numerical methods use discretization a have slow rate of convergence. Iterative methods are sensitive to initial conditions and in case of high nonlinearity they do not yield converged results. In perturbation methods small parameter is applied on the equation and hence cannot be applied for high nonlinear problems as they do not have small parameter. One of domain type methods is known as OHAM. This method is free from small parameter assumption and do not need the initial guess. The proposed method provides better accuracy at lower-order of approximations. Moreover the convergence domain can be easily adjusted. In this thesis OHAM is implemented for solution linear and nonlinear tenth order ODEs. Then its effectiveness and generalization is shown to a nonlinear family of PDEs, including Burger, Fisher, Burger’s–Huxley, Burger’s–Fisher, MEW and DGRLW equations. The results of the proposed method are compared with that of DTM, VIM, ADM, HAM and HPM, which reveal that OHAM is effective, simpler, easier and explicit. Apart from application to PDEs, OHAM is applied to couple system of PDEs. The coupled WBK, ALW, MB systems are used as test examples and results are compared with those obtained by HPM. OHAM is implemented to DDEs as well, and solution of MKdV lattice equation is presented for the illustration of proposed technique. The results are compared with HAM and HPM. In all cases the results obtained by OHAM are in close agreement with the exact solution and reveal high accuracy.