The aim of this paper is to extend the split-step idea for the solution of fractional partial differential equations. We consider the multidimensional nonlinear Schr"{o}dinger equation with the Riesz space fractional derivative and propose an efficient numerical algorithm to obtain it's approximate solutions. To this end, we first discretize the Riesz fractional derivative then apply the Crank-...

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations wi...

In this paper, we propose a finite difference method for the Riesz space fractional diffusion equations with delay and a nonlinear source term on a finite domain. The proposed method combines a time scheme based on the predictor-corrector method and the Crank-Nicolson scheme for the spatial discretization. The corresponding theoretical results including stability and convergence are provided. S...

Bio-heat transfer is the study of external or internal heat transfer in the biological body. In different therapeutic treatments especially in cancer treatment, heat is used to cure infected cells. The required temperature that will kill the infected cell should be known before starting the thermal treatment on human tissue. The useful ways to measure the temperature distribution on human tissu...

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficient on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the...

We analyze the discretization of an initial-boundary value problem for the cubic Schrödinger equation in one space dimension by a Crank–Nicolson–type finite difference scheme. We then linearize the corresponding equations at each time level by Newton’s method and discuss an iterative modification of the linearized scheme which requires solving linear systems with the same tridiagonal matrix. We...

This paper illustrates the use of the differentiation matrix technique for solving differential equations in finance. The technique provides a compact and unified formulation for a variety of discretisation and time-stepping algorithms for solving problems in one and two dimensions. Using differentiation matrix models, we compare time-stepping algorithms for option pricing computations and pres...