# 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:

- 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}

}