Numerical solutions of the differential equations having quadratic and cubic nonlinearities by q-homotopy analysis transform method

In this thesis deals with the solutions of differential equations having quadratic and cubic nonlinearity obtained by the q-homotopy analysis transform method (q-HATM). The q-HATM, combined by the q-homotopy analysis method and Laplace transform method, can be applied to a wide class of the partial and ordinary differential equations. In study, firstly, the usage areas and the historical process of q-HATM are examined. In addition, the connections between q-HATM, homotopy analysis method, q-Homotopy analysis method and Laplace transformation method are explained. Thereafter, the basic concepts of q-HATM, Ricatti differential equations, Fisher equations, Burgers-Fisher equations, Van der Pol-Duffing equations and Klein Gordon equations are mentioned. Then the method of applying the q-HATM for differential equations with having quadratic and cubic nonlinearity terms is explained. Through this method, convergent series solutions of Ricatti differential equations and Fisher equations having quadratic nonlinearity and numerical solutions of Burgers-Fisher equation and Van der Pol-Duffing equation having cubic nonlinearity are obtained. Finally, in order to reliability of the q-HATM, error analysis are performed and the results are shown and interpreted with graphs.