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 Cauchy problem for general time fractional diffusion equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1545-1559
Author(s):  
Chung-Sik Sin
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Behavior ◽  
Analysis Technique ◽  
The Cauchy Problem ◽  
Long Time ◽  
Time Fractional Diffusion Equation

Abstract In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.

scholarly journals REVISITING THE DERIVATION OF THE FRACTIONAL DIFFUSION EQUATION

Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 281-289 ◽  
Author(s):  
ENRICO SCALAS ◽  
RUDOLF GORENFLO ◽  
FRANCESCO MAINARDI ◽  
MARCO RABERTO
Keyword(s):  
Cauchy Problem ◽  
Limit Theorem ◽  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Diffusion Limit ◽  
Straightforward Application ◽  
The Cauchy Problem ◽  
Continuous Time Random Walks

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.

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.

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