Related papers: A novel second order finite difference discrete sc…
We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$\alpha \in (0,1)$. The basic idea of our scheme is based on local integration followed by linear…
The aim of this work is to provide the first strong convergence result of numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order…
We develop a fully discrete scheme for time-fractional diffusion equations by using a finite difference method in time and a finite element method in space. The fractional derivatives are used in Caputo sense. Stability and error estimates…
We begin with a treatment of the Caputo time-fractional diffusion equation, by using the Laplace transform, to obtain a Volterra intego-differential equation where we may examine the weakly singular nature of this convolution…
The discrete gradient structure and the positive definiteness of discrete fractional integrals or derivatives are fundamental to the numerical stability in long-time simulation of nonlinear integro-differential models. We build up a…
In this paper, we present a new fractal derivative with a nonsingular kernel and analyze its fundamental properties. The effectiveness of the proposed operator is illustrated through the study of economic models using both the Caputo…
In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The…
Adaptive second-order Crank-Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional Molecular Beam Epitaxial models with Caputo's derivative. Based on the piecewise…
The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in…
We present an efficient algorithm for the evaluation of the Caputo fractional derivative $_0^C\!D_t^\alpha f(t)$ of order $\alpha\in (0,1)$, which can be expressed as a convolution of $f'(t)$ with the kernel $t^{-\alpha}$. The algorithm is…
This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…
In the paper we deal with linear fractional control problems with constant delays in the state. Single-order systems with fractional derivative in Caputo sense of orders between 0 and 1 are considered. The aim is to introduce a new…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
In this paper we present a new type of fractional operator, the Caputo-Katugampola derivative. The Caputo and the Caputo-Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a…
We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…
In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify…
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference…
This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak…
In this paper, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We first derive an approximate evolution equation for the macroscopic…
When $0<\alpha<1$, the approximation for the Caputo derivative $$y^{(\alpha)}(x) = \frac{1}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^n \sigma_k^{(\alpha)} y(x-kh)+O\bigl(h^{2-\alpha}\bigr),$$ where $\sigma_0^{(\alpha)} = 1, \sigma_n^{(\alpha)} =…