Numerical solution of the multiterm time‐fractional diffusion equation based on reproducing kernel theory

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2016.y13024 ◽

2016 ◽

Vol 9 (3) ◽

pp. 337-357 ◽

Cited By ~ 19

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.

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Solving Time-Fractional Diffusion Equation: A Finite-Element Approach

Science Journal of University of Zakho ◽

10.25271/sjuoz.2021.9.1.791 ◽

2021 ◽

Vol 9 (1) ◽

pp. 38-42

Author(s):

Hussein J. Zekri

Keyword(s):

Finite Element ◽

Diffusion Equation ◽

Numerical Solution ◽

Uniform Space ◽

Absolute Error ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Element Approach ◽

Time Fractional Diffusion Equation ◽

The Stability

The numerical solution for a time-fractional diffusion equation supplemented with initial and boundary conditions is considered. The scheme is based on the Galerkin finite element method. The uniform space discretization is applied to study the stability of the solution of the problem within our approach. An analytically solvable example is presented to make a comparison between the exact solution and our numerical solution. By presenting the absolute error with different step-sizes and different values for time-fractional derivative, reliability and efficiency of our proposed numerical method is manifested.

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