*Correspondence: [email protected] 2School of Science, Jiangnan University, Lihu Road, Wuxi, 214122, China Full list of author information is available at the end of the article Abstract In the study of pattern formation in bi-stable systems, the extended Fisher-Kolmogorov (EFK) equation plays an important role. In this paper, a Fourier pseudo-spectral method for solving the EFK equation in ...

A new method is formulated and analyzed for the approximate solution of a twodimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit (ADI) method based on the Crank-Nicolson method combined with the L1-approximation of the time Caputo derivative of...

In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric Lévy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the t...

Abstract—A physical model for guiding the wave in photorefractive media is studied. Propagation of cos-Gaussian beam as the special cases of sinusoidal-Gaussian beams in photorefractive crystal is simulated numerically by the Crank-Nicolson method in one dimension. Results show that the beam profile deforms as the energy transfers from the center to the tails under propagation. This simulation ...

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...

We propose and analyze a linear stabilization of the Crank-Nicolson Leap-Frog (CNLF) method that removes all timestep / CFL conditions for stability and controls the unstable mode. It also increases the SPD part of the linear system to be solved at each time step. We give a proof of unconditional stability of the method as well as a proof of unconditional, asymptotic stability of both the stabl...

We propose and analyze a partitioned numerical method for the fully evolutionary Stokes-Darcy equations that model the coupling between surface and groundwater flows. The proposed method uncouples the surface from the groundwater flow by using the implicit-explicit combination of the Crank-Nicolson and Leapfrog methods for the discretization in time with added stabilization terms. We prove that...