scholarly journals A new fractional derivative operator and its application to diffusion equation

2021 ◽  
Author(s):  
Ruchi Sharma ◽  
Pranay Goswami ◽  
RAVI DUBEY ◽  
Fethi Belgacem
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Derivative Operator ◽  
Fractional Diffusion Equations

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

scholarly journals Maximum principle and its application for the time-fractional diffusion equations

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Yury Luchko
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
A Priori Estimates ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations ◽  
Generalized Time

AbstractIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.

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.

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

2017 ◽  
Vol 22 (4) ◽  
pp. 1028-1048 ◽  
Author(s):  
Yonggui Yan ◽  
Zhi-Zhong Sun ◽  
Jiwei Zhang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fast Evaluation ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Order Scheme ◽  
Speed Up

AbstractThe fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on theL2-1σformula proposed in [A. Alikhanov,J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

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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