scholarly journals Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain

2012 ◽  
Vol 389 (2) ◽  
pp. 1117-1127 ◽  
Author(s):  
H. Jiang ◽  
F. Liu ◽  
I. Turner ◽  
K. Burrage
Keyword(s):  
Analytical Solutions ◽  
Diffusion Equations ◽  
Finite Domain ◽  
Advection Diffusion ◽  
Time Space

scholarly journals A Finite Element Method for the Multiterm Time-Space Riesz Fractional Advection-Diffusion Equations in Finite Domain

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Jingjun Zhao ◽  
Jingyu Xiao ◽  
Yang Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Time Discretization ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Domain ◽  
Weak Formulation ◽  
Advection Diffusion ◽  
Time Space ◽  
Element Method

We present an effective finite element method (FEM) for the multiterm time-space Riesz fractional advection-diffusion equations (MT-TS-RFADEs). We obtain the weak formulation of MT-TS-RFADEs and prove the existence and uniqueness of weak solution by the Lax-Milgram theorem. For multiterm time discretization, we use the Diethelm fractional backward finite difference method based on quadrature. For spatial discretization, we show the details of an FEM for such MT-TS-RFADEs. Then, stability and convergence of such numerical method are proved, and some numerical examples are given to match well with the main conclusions.

Analytical solutions of the space–time fractional Telegraph and advection–diffusion equations

2018 ◽  
Vol 491 ◽  
pp. 810-819 ◽  
Author(s):  
Ashraf M. Tawfik ◽  
Horst Fichtner ◽  
Reinhard Schlickeiser ◽  
A. Elhanbaly
Keyword(s):  
Analytical Solutions ◽  
Space Time ◽  
Diffusion Equations ◽  
Advection Diffusion

Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Xiao-Li Ding ◽  
Juan J. Nieto
Keyword(s):  
Analytical Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Infinite Domain ◽  
Time Space ◽  
Special Cases ◽  
Telegraph Equations ◽  
Wave Problems ◽  
Fractional Reaction

AbstractWe consider the analytical solutions of multi-term time-space fractional reaction-diffusion equations on an infinite domain. The results are presented in a compact and elegant form in terms of the Mittag-Leffler functions. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, fractional wave problems, and fractional telegraph equations scattered in the literature can be derived as special cases of the results presented in this paper.

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