scholarly journals Averaged Control for Fractional ODEs and Fractional Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Darko Mitrovic ◽  
Andrej Novak ◽  
Tarik Uzunović
Keyword(s):  
Banach Space ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Fractional Type

We generalize results concerning averaged controllability on fractional type equations: system of fractional ODEs and the fractional diffusion equation. The proofs are accomplished by introducing appropriate Banach space in which we prove observability inequalities.

Second Order Accuracy Finite Difference Methods for Fractional Diffusion Equations

2013 ◽  
Author(s):  
Yuki Takeuchi ◽  
Reiji Suda
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Fractional Diffusion ◽  
Second Order ◽  
Fractional Diffusion Equation ◽  
Diffusion Equations ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Difference Methods

Finite difference methods for fractional differential equation are ever proposed. However, precise error orders have not been analyzed for the methods higher than first order accuracy. This paper proposes a few finite difference methods for fractional diffusion equations and shows our methods have second order accuracy under the conditions that the solution functions have higher order than second order at boundaries. In addition, we show that the accuracy may decrease in the case that the solution functions have lower order than second order at boundaries when we use second order accuracy scheme. In this paper, we treat schemes based on Grunwald-Letnikov definition and apply them to three kinds of fractional diffusion equations using Riemann-Liouville derivative operator including time-fractional diffusion equation, space-fractional diffusion equation and time-space-fractional diffusion equation. Finally, we show the simulation results which indicate that our methods are stable and have successfully second order accuracy under the assumed conditions.

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.

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.

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.

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.

scholarly journals From continuous time random walks to the generalized diffusion equation

2018 ◽  
Vol 21 (1) ◽  
pp. 10-28 ◽  
Author(s):  
Trifce Sandev ◽  
Ralf Metzler ◽  
Aleksei Chechkin
Keyword(s):  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Probability Distribution Functions ◽  
Special Cases ◽  
Continuous Time Random Walks ◽  
Random Walk Theory

AbstractWe obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

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