On the stable implicit finite differences approximation of diffusion equation with the time fractional derivative without singular kernel

2019 ◽  
Vol 13 (06) ◽  
pp. 2050111 ◽  
Author(s):  
Ali Taghavi ◽  
Afshin Babaei ◽  
Alireza Mohammadpour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Finite Differences ◽  
Numerical Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Finite Differences Method ◽  
Time Fractional Diffusion Equation ◽  
The Stability

In this paper, we give a numerical approximation to the Caputo–Fabrizio time fractional diffusion equation. The implicit finite differences method is applied to solve a time-fractional diffusion equation with this new fractional derivative. We present the stability and convergence analysis of the proposed numerical scheme. Some numerical problems will be presented to show the accuracy and effectiveness of the method.

scholarly journals The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kaouther Bouchama ◽  
Yacine Arioua ◽  
Abdelkrim Merzougui
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Numerical Approximation ◽  
Fractional Diffusion ◽  
Mathematical Induction ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Approximation Solution ◽  
Time Fractional Diffusion Equation

<p style='text-indent:20px;'>In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.</p>

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

2019 ◽  
Vol 37 (2) ◽  
pp. 730-752 ◽  
Author(s):  
Jufeng Wang ◽  
Fengxin Sun
Keyword(s):  
Diffusion Equation ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Shape Functions ◽  
Content Type ◽  
Time Fractional Diffusion Equation ◽  
The Stability

Purpose This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions. Design/methodology/approach In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly. Findings Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate. Originality/value This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.

scholarly journals A new compact alternating direction implicit method for solving two dimensional time fractional diffusion equation with Caputo-Fabrizio derivative

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3609-3626
Author(s):  
Mehran Taghipour ◽  
Hossein Aminikhah
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
Theoretical Considerations

In this paper, a new compact alternating direction implicit (ADI) difference scheme is proposed for the solution of two dimensional time fractional diffusion equation. Theoretical considerations are discussed. We show that the proposed method is fourth order accurate in space and two order accurate in time. The stability and convergence of the compact ADI method are presented by the Fourier analysis method. Numerical examples confirm the theoretical results and high accuracy of the proposed scheme.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

scholarly journals A novel series method for fractional diffusion equation within Caputo fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 695-699 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Wei-Ping Zhong ◽  
Xiao-Jun Yang
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Series Solution ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Series Method ◽  
Time Fractional Diffusion Equation ◽  
Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

A modified trigonometric cubic B-spline collocation technique for solving the time-fractional diffusion equation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Neeraj Dhiman ◽  
M.J. Huntul ◽  
Mohammad Tamsir
Keyword(s):  
Diffusion Equation ◽  
Numerical Technique ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Dirichlet Boundary Conditions ◽  
Unconditionally Stable ◽  
Content Type ◽  
B Spline ◽  
Time Fractional Diffusion Equation ◽  
The Stability

Purpose The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional diffusion equation (TFDE). The TFDE has numerous applications to model many real objects and processes. Design/methodology/approach The time-fractional derivative is used in the Caputo sense. A modification is made in trigonometric cubic B-spline (TCB) functions for handling the Dirichlet boundary conditions. The modified TCB functions have been used to discretize the space derivatives. The stability of the technique is also discussed. Findings The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. The stability analysis shows that the method is unconditionally stable. Furthermore, this technique is efficient and requires less storage. Originality/value The current work is novel for solving TFDE. This technique is unconditionally stable and gives better results than existing results (Ford et al., 2011; Sayevand et al., 2016; Ghanbari and Atangana, 2020).

Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients

2017 ◽  
Vol 18 (5) ◽  
pp. 385-393
Author(s):  
Zeting Liu ◽  
Shujuan Lü
Keyword(s):  
Diffusion Equation ◽  
Pseudospectral Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Initial Value ◽  
Fully Discrete ◽  
Time Fractional Diffusion Equation ◽  
Gauss Points

Abstract:We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis.

scholarly journals A Difference Method for Solving the Steklov Nonlocal Boundary Value Problem of Second Kind for the Time-Fractional Diffusion Equation

2017 ◽  
Vol 17 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Anatoly A. Alikhanov
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Test Problems ◽  
Difference Problem ◽  
Time Fractional Diffusion Equation

AbstractWe consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters α, β and γ. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes. The obtained results are supported by the numerical calculations carried out for some test problems.

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