## Tbilisi Mathematical Journal

- Tbilisi Math. J.
- Volume 10, Issue 1 (2017), 127-144.

### Time evolution of the approximate and stationary solutions of the Time-Fractional Forced-Damped-Wave equation

A. M. A. El-Sayed, E. A. Abdel-Rehim, and A. S. Hashem

#### Abstract

In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion advection forced wave) is given for different parameters. The common finite difference rules beside the backward Grünwald–Letnikov scheme are used to find the approximation solution of this model. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. We follow the waves till they reach their stationary waves. The Von-Neumann stability condition is also considered and discussed. Besides the simulation of the time evolution of the approximation solution of the classical and time-fractional model, the stationary solutions are also simulated. All the numerical results are compared for different values of time.

#### Article information

**Source**

Tbilisi Math. J., Volume 10, Issue 1 (2017), 127-144.

**Dates**

Received: 17 June 2016

Accepted: 8 July 2016

First available in Project Euclid: 26 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tbilisi/1527300022

**Digital Object Identifier**

doi:10.1515/tmj-2017-0008

**Mathematical Reviews number (MathSciNet)**

MR3610025

**Zentralblatt MATH identifier**

1364.35422

**Subjects**

Primary: 26A33: Fractional derivatives and integrals

Secondary: 35L05: Wave equation 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 60J60: Diffusion processes [See also 58J65] 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes 65N06: Finite difference methods 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

**Keywords**

approximation solution diffusion wave equation explicit scheme Grünwald-Letnikov scheme time fractional stability stationary solution simulation

#### Citation

El-Sayed, A. M. A.; Abdel-Rehim, E. A.; Hashem, A. S. Time evolution of the approximate and stationary solutions of the Time-Fractional Forced-Damped-Wave equation. Tbilisi Math. J. 10 (2017), no. 1, 127--144. doi:10.1515/tmj-2017-0008. https://projecteuclid.org/euclid.tbilisi/1527300022