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Title: Numerical Method for Fractional Advection-Diffusion Equation with Heredity

Abstract

We propose a method of construction of difference schemes for fractional partial differential equations with delay in time. For the fractional equation with two-sided diffusion, fractional transfer in time, and a functional aftereffect, we construct an implicit difference scheme. We use the shifted Grünwald–Letnikov formulas for the approximation of fractional derivatives with respect to spatial variables and the L1-algorithm for the approximation of fractional derivatives in time. Also we use piecewise constant interpolation and extrapolation by extending the discrete prehistory of the model in time. The algorithm is a fractional analog of a purely implicit method; on each time step, it is reduced to the solution of linear algebraic systems. We prove the stability of the method and find its order of convergence.

Authors:
 [1]
  1. Ural Federal University, Institute of Mathematics and Mechanics, Ural Branch of RAS (Russian Federation)
Publication Date:
OSTI Identifier:
22771275
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Sciences
Additional Journal Information:
Journal Volume: 230; Journal Issue: 5; Conference: International symposium on differential equations, Perm (Russian Federation), 17-18 May 2016; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ADVECTION; ALGORITHMS; APPROXIMATIONS; CONVERGENCE; DIFFUSION EQUATIONS; EXTRAPOLATION; GENETICS; INTERPOLATION; NUMERICAL ANALYSIS

Citation Formats

Pimenov, V. G., E-mail: V.G.Pimenov@urfu.ru. Numerical Method for Fractional Advection-Diffusion Equation with Heredity. United States: N. p., 2018. Web. doi:10.1007/S10958-018-3780-6.
Pimenov, V. G., E-mail: V.G.Pimenov@urfu.ru. Numerical Method for Fractional Advection-Diffusion Equation with Heredity. United States. doi:10.1007/S10958-018-3780-6.
Pimenov, V. G., E-mail: V.G.Pimenov@urfu.ru. Tue . "Numerical Method for Fractional Advection-Diffusion Equation with Heredity". United States. doi:10.1007/S10958-018-3780-6.
@article{osti_22771275,
title = {Numerical Method for Fractional Advection-Diffusion Equation with Heredity},
author = {Pimenov, V. G., E-mail: V.G.Pimenov@urfu.ru},
abstractNote = {We propose a method of construction of difference schemes for fractional partial differential equations with delay in time. For the fractional equation with two-sided diffusion, fractional transfer in time, and a functional aftereffect, we construct an implicit difference scheme. We use the shifted Grünwald–Letnikov formulas for the approximation of fractional derivatives with respect to spatial variables and the L1-algorithm for the approximation of fractional derivatives in time. Also we use piecewise constant interpolation and extrapolation by extending the discrete prehistory of the model in time. The algorithm is a fractional analog of a purely implicit method; on each time step, it is reduced to the solution of linear algebraic systems. We prove the stability of the method and find its order of convergence.},
doi = {10.1007/S10958-018-3780-6},
journal = {Journal of Mathematical Sciences},
issn = {1072-3374},
number = 5,
volume = 230,
place = {United States},
year = {2018},
month = {5}
}