An eighth order frozen Jacobian iterative method for solving nonlinear IVPs and BVPs

Abstract

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems. 379 ordinary differential equations. They used implicit Runge-Kutta method to solve the system of ordinary differential equations and obtained highly accurate numerical solutions, for four different kinds of nonlinear 1+1 Schrödinger equations. In another article Bhrawy et al. [5] solved the nonlinear reaction-diffusion equations by using J-GL-C method. In the solution of the complex generalized Zakharov system of equation [4], the J-GL-C method is also the method of discretization. The further application of pseudospectral collocation techniques for solving nonlinear IVPs and BVPs can be found in [6,8,9,17] and references therein. The J-GL-C method is a parametric pseudospectral collocation method. By selecting different values of the parameters, we can get different pseudospectral collocation methods. The Legendre, Chebyshev, and Gegenbauer pseudospectral collocation methods are special cases of J-GL-C method [16].The Jacobi polynomials are the eigenfunctions of a singular Strum-Liouville problem [16] (1 − x 2 )σ (x) + (θ − φ + (θ + φ + 2)x)σ (x) + n(n + θ + φ + 1)σ(x) = 0 .The following recurrence relation produces the Jacobi polynomials J