scholarly journals Fundamental solutions to advection–diffusion equation with time-fractional Caputo–Fabrizio derivative

2017 ◽  
Vol 73 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Itrat Abbas Mirza ◽  
Dumitru Vieru
Keyword(s):  
Diffusion Equation ◽  
Fundamental Solutions ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

Analysis of advective–diffusive transport phenomena modelled via non-singular Mittag-Leffler kernel

2019 ◽  
Vol 14 (3) ◽  
pp. 309
Author(s):  
Derya Avci ◽  
Aylіn Yetіm
Keyword(s):  
Diffusion Equation ◽  
Fundamental Solutions ◽  
Diffusive Transport ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Main Motivation ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Fourier Transform Methods ◽  
Computational Processes

In this study, a linear advection–diffusion equation described by Atangana–Baleanu derivative with non-singular Mittag-Leffler kernel is considered. The Cauchy, Dirichlet and source problems are formulated on the half-line. The main motivation of this work is to find the fundamental solutions of prescribed problems. For this purpose, Laplace transform method with respect to time t and sine/cosine-Fourier transform methods with respect to spatial coordinate x are applied. It is remarkable that the obtained results are quite similar to the existing fundamental solutions of advection–diffusion equation with time-Caputo fractional derivative. Although the results are mathematically similar in both formulations, the AB derivative is a non-singular operator and provides a significant advantage in the computational processes. Therefore, it is preferable to replace the Caputo derivative in modelling such diffusive transports.

scholarly journals Fundamental Solutions to Time-Fractional Advection Diffusion Equation in a Case of Two Space Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. Z. Povstenko
Keyword(s):  
Diffusion Equation ◽  
Half Plane ◽  
Bessel Functions ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Fundamental Solutions ◽  
Numerical Results ◽  
Dirichlet Problems ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

The fundamental solutions to time-fractional advection diffusion equation in a plane and a half-plane are obtained using the Laplace integral transform with respect to timetand the Fourier transforms with respect to the space coordinatesxandy. The Cauchy, source, and Dirichlet problems are investigated. The solutions are expressed in terms of integrals of Bessel functions combined with Mittag-Leffler functions. Numerical results are illustrated graphically.

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