Tbilisi Mathematical Journal

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

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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


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References

  • E. A. Abdel-Rehim, Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks ( Mensch&Buch Verlag, 2004), ISBN 3–89820–736–6 Available at http://www.diss.fu-berlin.de/2004/168/index.html.
  • E. A. Abdel-Rehim, Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations, Scientific Research: Appl. Math. 4 (2013), 1427–1440.
  • E. A. Abdel-Rehim, Implicit difference scheme of the space-time fractional advection diffusion equation, J. Fract. Calc Appl. Anal. 18, No. 6 (2015), 1252–1276.
  • O. P. Agrawal, A general solution for the fourth-order fractional diffusion-wave equation, J. Fract. Calc. Appl. Anal. 3, No. 1 (2000), 1–12.
  • O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, J. Nonlinear Dynam. 29 (2002), 145–155.
  • R. C. Cascaval, E. C. Eckstein, C. L. Forta and J. A. Goldstein, Fractional Telegraph Equations, J. Math. Anal. Appl. 276, No. 1 (2002), 145–159.
  • W. Chen, S. Holm, Modified Szabós wave equation models for lossy media obeying frequency power–law attenuation, J. Acoust. Soc. Am. 116 (2004), 2742–2750.
  • D. Constantinescu and M. Stoicescu, Fractal dynamics as long range memory modelling techniques, J. Physics. Auc. 21 (2011), 114–120.
  • M. Caputo, Linear models of dissipation whose Q is almost independent II, J. Geophys. Royal Astronom. Soc. 13, No. 5 (1967), 529–539.
  • F. A. Duck, Physical Properties of Tissue: A Comperhensive Reference Book, Academic Press, Boston, 1990.
  • A. M. A. Elsayed, Fractional-order diffusion and wave equation, Int. J. Theor. Phys. 35 (1996), 311–322.
  • R. Gorenflo, A. Iskenderov and Y. Luchko, Mapping Between Solutions of Fractional Wave Equations, J. Fract. Calc. Appl. Anal. 3 (2000), 75–86.
  • R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time-fractional diffusion: a discrete random walk approach, J. Nonlinear Dynam. 29 (2002), 129–143.
  • R. Gorenflo and E. A. Abdel-Rehim, Discrete models of time-fractional diffusion in a potential well, J. Fract. Calc. Appl. Anal. 8 (2005), 173–200.
  • M. A. E. Herzallah, A. M. A. Elsayed and D. Baleanu, On the fractional-order diffusion-wave process, Rom. J. Phys. 55 No. 3–4 (2010), 274–284.
  • J. K. Kelly, R. J. McGough and M. M. Meerschaert, Analytical time-domain Green's functions for power-law media, J. Acoust. Soc. Am. 124 (2008), 2861–2872.
  • V. Keyantuo, C. Lizama and M. Warma, Asymptotic behavior of fractional order semilinear evolution equations, J. Differ. Integral Equ. 26 No. 7–8 (2013), 757–780.
  • A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, the Netherlands, 2006.
  • P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, J. Comm. Pure Appl. Math. IX (1956), 267–293.
  • M. Liebler, S. Ginter, T. Dreyer and R. E. Riedlinger, Full wave modeling of therapeutic ultrasound: Efficient time-domaint. implemetation of the frequency power-law attenuation, J. Acoust. Soc. Am. 116 (2004), 2742–2750.
  • C. Lizama, Solutions of two-term fractional order differential equations with nonlocal initial conditions, Electron. J. Qual. Theory Differ. Equ. 82 (2012), 1–9.
  • F. Mainardi, P. Paradisi, Model of diffusive waves in viscoelasticity based on fractional calculus, in Proceedings of the IEEE Conference on Decision and Control, 5, O. R. Gonzales, IEEE, New York, 1997, pp. 4961–4966.
  • Y. Povstenk, Signaling problem for time-fractional diffusion-wave equation in a half-plane, J. Fract. Calc. Appl. Anal. 11 No. 3 (2008), 329–352.
  • H. M. Srivastava, R. K. Raina and X.–J. Yang, Special Functions in a Fractional Calculus and Related Fractional Differential Equations, Word Scientific–Imperial College Press, 2016.
  • M. Stojanovic, Numerical method for solving diffusion wave phenomena, J. Comp. Appl. Math. 2335 (2011) 3121–3137.
  • T. L. Szabo, Time domain wave equation for lossy media obeying a frequency power law, J. Acoust. Soc. Am. 96 (1994), 491–500.
  • Jin-Liang Wang and Hui-Feng Li, Surpassing the fractional derivative: Concept of the memory-dependent derivative, J. Comp. Math. with Appl. 62 (2011), 1562–1567.
  • X.-J. Yang, D. Baleanu and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications, Academic Press (Elsevier Science Publishers), Amsterdam, Heidelberg, London and New York, 2016.