Related papers: Finite difference approximations for the first-ord…
In this paper, we propose numerical scheme for the Riesz space fractional advection-dispersion equations with delay (RFADED). Firstly, analytical solution for RFADED in terms of the functions of Mittag-Leffler type is derived. Secondly, the…
Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine…
We give sufficient conditions under which the convergence of finite difference approximations in the space variable of the solution to the Cauchy problem for linear stochastic PDEs of parabolic type can be accelerated to any given order of…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for…
In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$…
In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate…
In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of…
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…
This paper presents a concurrent global-local numerical method for solving multiscale parabolic equations in divergence form. The proposed method employs hybrid coefficient to provide accurate macroscopic information while preserving…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local $L^1$-error between the exact and numerical solutions is $\mathcal{O}(\Delta x^{2/(19+d)})$,…
This paper is dedicated to the construction of high-order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations…
Based on the weighted and shifted Gr\"{u}nwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the…
This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional…
Based on our recent results, in this paper, a compact finite difference scheme is derived for a time fractional differential equation subject to the Neumann boundary conditions. The proposed scheme is second order accurate in time and…
The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and…
In this article, firstly we develop a method for a type of difference equations, applicable to solve approximately a class of first order ordinary differential equation systems. In a second step, we apply the results obtained to solve a…