scholarly journals Two equivalent Stefan’s problems for the time fractional diffusion equation

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Sabrina Roscani ◽  
Eduardo Marcus
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Classical Solutions ◽  
Constant Condition ◽  
Flux Condition ◽  
Time Fractional Diffusion Equation ◽  
Fractional Equation

AbstractTwo Stefan’s problems for the diffusion fractional equation are solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. The first one has a constant condition on x = 0 and the second presents a flux condition T x(0,t) = q/t α/2. An equivalence between these problems is proved and the convergence to the classical solutions is analyzed when α ↗ 1 recovering the heat equation with its respective Stefan’s condition.

scholarly journals A new equivalence of Stefan’s problems for the time fractional diffusion equation

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Sabrina Roscani ◽  
Eduardo Marcus
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Classical Solutions ◽  
Constant Condition ◽  
Convective Condition ◽  
Flux Condition ◽  
Time Fractional Diffusion Equation

AbstractA fractional Stefan’s problem with a boundary convective condition is solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. Then an equivalence with other two fractional Stefan’s problems (the first one with a constant condition on x = 0 and the second with a flux condition) is proved and the convergence to the classical solutions is analyzed when α ↗ 1 recovering the heat equation with its respective Stefan’s condition.

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.

On the Approximate Solution of Caputo-Riesz-Feller Fractional Diffusion Equation

2020 ◽  
pp. 224-244
Author(s):  
Samir Shamseldeen ◽  
Ahmed Elsaid ◽  
Seham Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Fractional Time Derivative ◽  
Complex Exponential ◽  
Fractional Equation

In this work, a space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative is introduced. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. Also, a very useful Riesz-Feller fractional derivative is proved; the property is essential in applying iterative methods specially for complex exponential and/or real trigonometric functions. The analytic series solution of the problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to highlight the effect of changing the fractional derivative parameters on the behavior of the obtained solutions. The results in this work are originally extracted from the author's work.

scholarly journals Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammed Syam ◽  
Mohammed Al-Refai
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Path Following ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Legendre Functions ◽  
Time Fractional Diffusion Equation ◽  
Path Following Methods ◽  
Generalized Time

A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented.

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 Semianalytic Solution of Space-Time Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
A. Elsaid ◽  
S. Shamseldeen ◽  
S. Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Time Fractional Diffusion Equation ◽  
Fractional Time Derivative

We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.

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>

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