This paper discusses the study of optical solitons that are modeled by Riesz fractional Chen-Lee-Liu model, one versions famous nonlinear Schrödinger equation. model is solved assistance consecutive spectral collocation technique with two independent approaches. The first approach spatial variable, while other temporal variable. It concluded method current far more efficient and credible for pr...

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the δ-function. For the approximation of the δ-function, the direct projection method is used that has been proposed i...

We consider a linear full elliptic second order partial differential equation in a d-dimensional domain, d ≥ 1, approximated by isogeometric collocation methods based on uniform B-splines of degrees p := (p1, . . . , pd), pj ≥ 2, j = 1, . . . , d. We give a construction of the inherently non-symmetric matrices arising from this approximation technique and we perform an analysis of their spectra...

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier–Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the L2 and H1 norms further leads to the convergence of th...

In this paper, we propose an iterative spectral method for solving differential equations with initial values on large intervals. In the proposed method, we first extend the Legendre wavelet suitable for large intervals, and then the Legendre-Guass collocation points of the Legendre wavelet are derived. Using this strategy, the iterative spectral method converts the differential equation to a s...

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function...

We present a semi-Lagrangian method for integrating the three-dimensional incompressible NavierStokes equations. We develop stable schemes of second-order accuracy in time and spectral accuracy in space. Specifically, we employ a spectral element (Jacobi) expansion in one direction and Fourier collocation in the other two directions. We demonstrate exponential convergence for this method, and i...

Meshless collocation methods based on radial basis functions lead to structured linear systems which, for equispaced grid points, have almost a multilevel Toeplitz structure. In particular, if we consider partial differential equations (PDEs) in two dimensions then we find almost (up to a “low-rank” correction given by the boundary conditions) two-level Toeplitz matrices, i.e., block Toeplitz w...