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 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.

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 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 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.

scholarly journals Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Y. J. Choi ◽  
S. K. Chung
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Source Term ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Backward Euler ◽  
Space Fractional Diffusion Equation

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained asO(k+hγ˜), whereγ˜is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

scholarly journals Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution

2021 ◽  
Vol 2 (1) ◽  
pp. 60-75
Author(s):  
Ndolane Sene
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Integral Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Double Integral ◽  
Uniqueness Of The Solution ◽  
The Impact ◽  
The Heat Balance

In this paper, we propose the approximate solution of the fractional diffusion equation described by a non-singular fractional derivative. We use the Atangana-Baleanu-Caputo fractional derivative in our studies. The integral balance methods as the heat balance integral method introduced by Goodman and the double integral method developed by Hristov have been used for getting the approximate solution. In this paper, the existence and uniqueness of the solution of the fractional diffusion equation have been provided. We analyze the impact of the fractional operator in the diffusion process. We represent graphically the approximate solution of the fractional diffusion equation.

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