scholarly journals Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval

2021 ◽  
Vol 86 (3) ◽  
Author(s):  
Xiangcheng Zheng ◽  
V. J. Ervin ◽  
Hong Wang
Keyword(s):  
Approximation Method ◽  
Fractional Diffusion ◽  
Spectral Approximation ◽  
Reaction Equation ◽  
Bounded Interval

On the fractional diffusion-advection-reaction equation in ℝ

2019 ◽  
Vol 22 (4) ◽  
pp. 1039-1062 ◽  
Author(s):  
Victor Ginting ◽  
Yulong Li
Keyword(s):  
Sobolev Space ◽  
Anomalous Diffusion ◽  
Differential Operators ◽  
Weak Sense ◽  
Fractional Diffusion ◽  
Reaction Equation ◽  
State Behavior ◽  
Harmonic Analyses ◽  
Steady State Behavior

Abstract We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

SOLVING AN INCOMPLETE DATA INVERSE PROBLEM BY A PSEUDO-SPECTRAL APPROXIMATION METHOD WITH A NON-STANDARD APPROACH

2019 ◽  
Vol 18 (2) ◽  
pp. 9-21
Author(s):  
Abani Maïdaoua Ali ◽  
Dia Bassirou ◽  
Diop Oulimata ◽  
Sembene Ama Diop Niang ◽  
Mampassi Benjamin
Keyword(s):  
Inverse Problem ◽  
Approximation Method ◽  
Incomplete Data ◽  
Standard Approach ◽  
Spectral Approximation ◽  
Pseudo Spectral

Mass-conserving tempered fractional diffusion in a bounded interval

2019 ◽  
Vol 22 (6) ◽  
pp. 1561-1595 ◽  
Author(s):  
Anna Lischke ◽  
James F. Kelly ◽  
Mark M. Meerschaert
Keyword(s):  
Numerical Solutions ◽  
Mass Conservation ◽  
Fractional Diffusion ◽  
Absorbing Boundary Conditions ◽  
Late Time ◽  
Conditional Stability ◽  
Absorbing Boundary ◽  
Bounded Interval ◽  
Reflecting Boundaries ◽  
Euler Methods

Abstract Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.

scholarly journals Leapfrog/Finite Element Method for Fractional Diffusion Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhengang Zhao ◽  
Yunying Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Element Analysis ◽  
Bounded Interval ◽  
Fully Discrete ◽  
Leapfrog Scheme ◽  
Element Method

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove anL2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.

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