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A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion
Author(s) -
Jani Mostafa,
Babolian Esmail,
Bhatta Dambaru
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6908
Subject(s) - petrov–galerkin method , mathematics , discretization , spectral method , anomalous diffusion , galerkin method , basis function , spectral element method , numerical analysis , convolution (computer science) , convergence (economics) , stability (learning theory) , mathematical analysis , quadrature (astronomy) , finite element method , computer science , physics , knowledge management , innovation diffusion , extended finite element method , machine learning , artificial neural network , optics , economics , thermodynamics , economic growth
Anomalous diffusion problems are used to describe the evolution of particle's motion in crowded environments with many applications, such as modeling the intracellular transport and disordered media. In the present paper, we develop a Petrov–Galerkin spectral method for the fourth‐order anomalous fractional diffusion equations. For the dimension reduction, we use a discretization in time by the convolution quadrature. We then introduce the basis sets for the trial‐and‐test spaces using modal Bernstein basis functions with a presentation of the method in a weak spectral formulation along with a discussion of the structure of the resulting systems, the convergence, and stability of the proposed method. The theoretical results are supported by illustrating some numerical examples.