1,888 research outputs found
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
Effects of fractional mass transfer and chemical reaction on MHD flow in a heterogeneous porous medium
This paper presents a study on space fractional anomalous convective-diffusion and chemical reaction in the magneto-hydrodynamic fluid over an unsteady stretching sheet. The fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium. A novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level. Numerical solutions are established by an implicit finite difference scheme. The stability and convergence of the method are analyzed. Results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall. There is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter. Moreover, the numerical solutions are compared with exact solutions and good agreement has been observed.</p
Two different fractional Stefan problems which are convergent to the same classical Stefan problem
Two fractional Stefan problems are considered by using Riemann-Liouville and
Caputo derivatives of order such that in the limit case
() both problems coincide with the same classical Stefan problem.
For the one and the other problem, explicit solutions in terms of the Wright
functions are presented. We prove that these solutions are different even
though they converge, when , to the same classical solution.
This result also shows that some limits are not commutative when fractional
derivatives are used.Comment: 14 pages, 1 figur
Robust stability of thermal control systems with uncertain parameters: The graphical analysis examples
This paper is intended to present the investigation of robust stability for integer order or fractional order feedback control loops affected by parametric uncertainty and time-delay(s) with special emphasis on the thermal control systems. The applied graphical method is based on the numerical calculations of the value sets and the zero exclusion condition. Three robust stability examples inspired by control of the real-world thermal processes are used for demonstration of the technique applicability. Namely, the work deals with the analysis of a shell-and-tube heat exchanger which was identified as the (integer order) time-delay model with parametric uncertainty, a heat transfer process modeled as the fractional order time-delay plant with parametric uncertainty, and a heating–cooling system with a heat exchanger described by the anisochronic model with internal delays and parametric uncertainty. © 2017 Elsevier LtdEuropean Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [LO1303 (MSMT-7778/2014)
An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator
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