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.

A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain

2019 ◽  
Vol 27 (5) ◽  
pp. 609-621 ◽  
Author(s):  
Fan Yang ◽  
Ni Wang ◽  
Xiao-Xiao Li ◽  
Can-Yun Huang
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Symmetric Domain ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Spherically Symmetric ◽  
Initial Value ◽  
Time Fractional Diffusion Equation

Abstract In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.

A Time-Fractional Diffusion Equation with Generalized Memory Kernel in Differential and Difference Settings with Smooth Solutions

2017 ◽  
Vol 17 (4) ◽  
pp. 647-660 ◽  
Author(s):  
Anatoly A. Alikhanov
Keyword(s):  
Diffusion Equation ◽  
Approximation Error ◽  
A Priori Estimate ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Fractional Diffusion Equation ◽  
Test Problems ◽  
Memory Kernel ◽  
Differential Problem ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, an a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the Caputo fractional derivative with generalized memory kernel (L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the {(2-\alpha)}-th order in time for the generalized time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid {L_{2}}-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.

scholarly journals An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

2021 ◽  
Vol 7 (2) ◽  
pp. 2370-2392
Author(s):  
Fouad Mohammad Salama ◽  
◽  
Nur Nadiah Abd Hamid ◽  
Norhashidah Hj. Mohd Ali ◽  
Umair Ali ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Iterative Method ◽  
Method Comparison ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Matrix Analysis ◽  
Stability And Convergence ◽  
Two Space Dimensions ◽  
Time Fractional Diffusion Equation

<abstract><p>In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.</p></abstract>

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 Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

2016 ◽  
Vol 9 (3) ◽  
pp. 337-357 ◽  
Author(s):  
Gongsheng Li ◽  
Chunlong Sun ◽  
Xianzheng Jia ◽  
Dianhu Du
Keyword(s):  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Upper And Lower Bounds ◽  
Stability And Convergence ◽  
Finite Domain ◽  
Time Space ◽  
Time Fractional Diffusion Equation

AbstractThis paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.

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.

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