A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

2020 ◽  
Vol 37 (1) ◽  
pp. 643-673
Author(s):  
Qinxu Ding ◽  
Patricia J. Y. Wong
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Generalized Fractional Derivative

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.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

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.

Sensitivity of the pressure distribution to the fractional order α in the fractional diffusion equation

2015 ◽  
Vol 93 (1) ◽  
pp. 18-36 ◽  
Author(s):  
Nadeem A. Malik ◽  
R.A. Ghanam ◽  
S. Al-Homidan
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Pressure Distribution ◽  
Fractional Diffusion ◽  
Linear Increase ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Rock Formations ◽  
Rate Of Increase ◽  
Porous Reservoir

In reservoir engineering, an oil reservoir is commonly modeled using Darcy’s diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≤ α < 1 (e.g., K ∝ (1 – α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t) – pi)/pi in the reservoir for large times, where pi = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1 – α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α.

scholarly journals Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Operators ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Unique Weak Solution ◽  
Bounded Lipschitz Domain

AbstractIn this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

scholarly journals An Entropy Paradox Free Fractional Diffusion Equation

2021 ◽  
Vol 5 (4) ◽  
pp. 236
Author(s):  
Manuel Duarte Ortigueira
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Stable Distributions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Extreme Wave ◽  
Wave Regime ◽  
New Formulation ◽  
Different Order ◽  
New Look

A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formulae used for the stable distributions. The entropy paradox problem was studied and clarified through the Rényi entropy: in the extreme wave regime the entropy is −∞. In passing, Tsallis and Rényi entropies for stable distributions are introduced and exemplified.

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.

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