A hybrid numerical method based on the generalized pseudospectral method for solving nonlinear differential equations

In this paper, a hybrid numerical method using generalized pseudospectral and Newton-Kantorovich quasilinearization methods is presented to solve nonlinear differential equations. Initially, generalized Lagrange functions as basic functions are introduced and then derivative operational matrices for these functions are presented. Then using these new functions, the generalized pseudospectral method is constructed as a numerical method. Finally, this method and the Newton-Kantorovich quasilinearization method are combined to produce an efficient method. Because of the use of derivative operating matrices and the conversion of any nonlinear differential equation into sequences of linear differential equations, the implementation of this method does not require mathematically to calculate the derivative and the computational costs are also reduced. To illustrate the efficiency, accuracy, and convergence of the method, the proposed method is implemented on two famous equations and the results are compared with other methods.