Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 6
Author(s):  
Mohammed Hamed Alshbool ◽  
Osman Isik ◽  
Ishak Hashim
Keyword(s):  
Diffusion Equation ◽  
Series Solution ◽  
Bernstein Polynomial ◽  
Absolute Error ◽  
Fractional Diffusion ◽  
Estimation Methods ◽  
Polynomial Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Diffusion Equations ◽  
Residual Correction

In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems in fractional partial differential equations. To get the FBSS, we first convert all terms in the problem to matrix forms. Then, the fundamental matrix equation is obtained and thus, the solution is obtained. Two error estimation methods based on a residual correction procedure and the consecutive approximations are incorporated to find the estimate and bound of the absolute error. The perturbation and stability analysis of the method is given. We apply the method to some illustrative examples. The numerical results are compared with the exact solutions and known second-order methods. The outcomes of the numerical examples are very encouraging and show that the FBSS is highly useful in solving fractional partial problems. The results show the accuracy and effectiveness of the method.

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 Existence of Positive Solution to the Cauchy Problem for a Fractional Diffusion Equation with a Singular Nonlinearity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Ailing Shi ◽  
Shuqin Zhang
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Positive Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Successive Approximation Method ◽  
Fractional Diffusion Equations ◽  
Singular Nonlinearity ◽  
Analysis Technique ◽  
The Cauchy Problem

Fractional diffusion equations describe an anomalous diffusion on fractals. In this paper, by means of the successive approximation method and other analysis technique, we present a local positive solution to Cauchy problem for a fractional diffusion equation with singular nonlinearity. The fractional derivative is described in the Caputo sense.

scholarly journals An Implementation Solution for Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Nicolas Bertrand ◽  
Jocelyn Sabatier ◽  
Olivier Briat ◽  
Jean-Michel Vinassa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Differentiation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Diffusion Equations ◽  
Electrochemical Interfaces ◽  
And Diffusion

The link between fractional differentiation and diffusion equation is used in this paper to propose a solution for the implementation of fractional diffusion equations. These equations permit us to take into account species anomalous diffusion at electrochemical interfaces, thus permitting an accurate modeling of batteries, ultracapacitors, and fuel cells. However, fractional diffusion equations are not addressed in most commercial software dedicated to partial differential equations simulation. The proposed solution is evaluated in an example.

A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations

2014 ◽  
Vol 875-877 ◽  
pp. 781-785 ◽  
Author(s):  
Jun Ying Cao ◽  
Chuan Ju Xu ◽  
Zi Qiang Wang
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Polynomial Degree ◽  
Fractional Diffusion Equations ◽  
Spectral Approximations ◽  
High Order Finite Difference

In this paper, we consider the numerical solution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order α, with 03-α+N-m) , where Δt,N and m are the time step size, the polynomial degree and the regularity of the exact solution, respectively.

scholarly journals Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
A. Elsaid ◽  
M. S. Abdel Latif ◽  
M. Maneea
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Series Solution ◽  
Error Function ◽  
Fractional Diffusion ◽  
Similarity Solutions ◽  
Fractional Diffusion Equation ◽  
Power Series Solution ◽  
Time Fractional Diffusion Equation ◽  
Similarity Method

Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval(0,1]and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.

Numerical Algorithm of the Fractional Diffusion Equation Based on Sinc-Chebyshev Collocation

2014 ◽  
Vol 926-930 ◽  
pp. 3105-3108
Author(s):  
Zhi Mao ◽  
Ting Ting Wang
Keyword(s):  
Diffusion Equation ◽  
Numerical Algorithm ◽  
Variable Coefficient ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Diffusion Equations ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

Fractional diffusion equations have recently been applied in various area of engineering. In this paper, a new numerical algorithm for solving the fractional diffusion equations with a variable coefficient is proposed. Based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized respectively, the problem is reduced to the solution of a system of linear algebraic equations. The procedure is tested and the efficiency of the proposed algorithm is confirmed through the numerical example.

scholarly journals Identification of points sources via time fractional diffusion equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6189-6201 ◽  
Author(s):  
A. Ghanmi ◽  
R. Mdimagh ◽  
I.B. Saad
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Cauchy Data ◽  
Diffusion Equations ◽  
Lipschitz Stability ◽  
Single Measurement ◽  
Fractional Diffusion Equations ◽  
Time Fractional Diffusion Equation

This article investigates the source identification in the fractional diffusion equations, by performing a single measurement of the Cauchy data on the accessible boundary. The main results of this work consist in giving an identifiability result and establishing a local Lipschitz stability result. To solve the inverse problem of identifying fractional sources from such observations, a non-iterative algebraical method based on the Reciprocity Gap functional is proposed.

scholarly journals Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data

2020 ◽  
Vol 20 (3) ◽  
pp. 573-589 ◽  
Author(s):  
Xiangcheng Zheng ◽  
Vincent J. Ervin ◽  
Hong Wang
Keyword(s):  
Diffusion Equation ◽  
Error Estimates ◽  
Variable Coefficient ◽  
Approximation Scheme ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusivity Coefficient ◽  
Fractional Diffusion Equations ◽  
Equation Error ◽  
Change Of Variable

AbstractIn this article, we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post-processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Numerical experiments are presented to test the performance of the proposed method.

scholarly journals An efficient fractional polynomial method for space fractional diffusion equations

2018 ◽  
Vol 9 (4) ◽  
pp. 767-776 ◽  
Author(s):  
K. Krishnaveni ◽  
K. Kannan ◽  
S. Raja Balachandar ◽  
S.G. Venkatesh
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Polynomial Method ◽  
Fractional Diffusion Equations ◽  
Fractional Polynomial

HOW DOES THE DIFFUSION EQUATION ON FRACTALS LOOK?

Fractals ◽  
2004 ◽  
Vol 12 (02) ◽  
pp. 149-156 ◽  
Author(s):  
H. EDUARDO ROMAN
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Satisfactory Answer

Different forms of diffusion equations on fractals proposed in the literature are reviewed and critically discussed. Variants of the known fractional diffusion equations are suggested here and worked out analytically. On the basis of these results we conclude that the quest: "what is the form of the diffusion equation on fractals," is still open, but we are possibly close to obtaining a satisfactory answer.

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