2020/10/03 by Stanislav Harizanov, Harizanov, Stanislav, Raytcho Lazarov +3 · 1 citation
Computer Science · Mathematics · #35R11 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2010.02717
openalex publication_date 2020/10/03 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
The survey is devoted to numerical solution of the fractional equation\nA^\α u=f, 0 < \α <1, where A is a symmetric positive definite\noperator corresponding to a second order elliptic boundary value problem in a\nbounded domain \Ω in mathbb Rd. The operator fractional power is a\nnon-local operator and is defined through the spectrum. Due to growing interest\nand demand in applications of sub-diffusion models to physics and engineering,\nin the last decade, several numerical approaches have been proposed, studied,\nand tested. We consider discretizations of the elliptic operator A by using\nan N-dimensional finite element space Vh or finite differences over a\nuniform mesh with N grid points.\n The numerical solution of this equation is based on the following three\nequivalent representations of the solution: (1) Dunford-Taylor integral formula\n(or its equivalent Balakrishnan formula), (2) extension of the a second order\nelliptic problem in \Ω \× (0,\∞)\⊂ mathbb Rd+1 (with a\nlocal operator) or as a pseudo-parabolic equation in the cylinder (x,t) \∈\n\Ω \× (0,1) , (3) spectral representation and the best uniform\nrational approximation (BURA) of z^\α on [0,1]. Though substantially\ndifferent in origin and their analysis, these methods can be interpreted as\nsome rational approximation of A-\α. In this paper we present the main\nideas of these methods and the corresponding algorithms, discuss their\naccuracy, computational complexity and compare their efficiency and robustness.\n