Analysis of the Difference Scheme of Wave Equation Equivalent with Fractional Differentiation Operator

V. D. Beybalaev ; A. Z. Yakubov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2014), pp. 125-133.

Résumé

Analysis of the difference scheme of boundary-value problem for the wave equation analogue is in the paper. Explicit and implicit difference schemes for numerical solution of the Caputo initial boundary-value problem of the analogue for the wave equation with fractional differentiation operator are invest gated, and the criteria of these difference schemes sustainability have been proved by the harmonic Fourier method. Estimations for Eigen values of the operator transition from one time layer to another are obtained. Computational experiment on the analysis of the given difference scheme has been performed on the basis of example graphs of the numerical solution of the boundary-value problem for the wave equation with the operator of fractional differentiation having different values of parameters of fractional differentiation $\alpha$ and $\beta$ have been built. Change of the period of fluctuations upon transition to a fractional derivative is established. On an example it is shown that parameters $\alpha$ and $\beta$ become managing directors.

Détail

Mots-clés : wave equation analogue
Keywords: Caputo fractional partial derivative, difference scheme.

Bibliographie

[1] Yu. I. Babenko, Metod drobnogo differentsirovaniya v prikladnykh zadachakh teorii teplomassoobmena [Method of Fractional Differentiation in Applied Problems of the Theory of Heat and Mass Transfer], Professional, St. Petersburg, 2009, 584 pp. (In Russian)

[2] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Jon Wiley Sons. Inc., New York, 1993, xiii+366 pp. | MR | Zbl

[3] I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, 1999, xxiv+340 pp. | MR | Zbl

[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 204, xv+523 pp. | MR | Zbl

[5] V. V. Vasilyev, L. O. Simak, Drobnoe ischislenie i approksimatsionnye metody v modelirovanii dinamicheskikh sistem [Fractional calculus and approximation methods for modeling of dynamic systems], NAN Ukraine, Kiev, 2008, 256 pp. (In Russian)

[6] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg, 2011, ix+612 pp. | DOI | MR | Zbl

[7] V. V. Uchaikin, Metod drobnykh proizvodnykh [The method of fractional derivatives], Artichoke Publ., Ulyanovsk, 2008, 512 pp. (In Russian)

[8] A. M. Nakhushev, Uravneniya matematicheskoy biologii [Equations of mathematical biology], Vysshaya Shkola, Moscow, 1995, 301 pp. (In Russian) | Zbl

[9] A. M. Nakhushev, Drobnoe ischislenie i ego primenenie [Fractional calculus and its applications], Fizmatlit, Moscow, 2003, 271 pp. (In Russian) | Zbl

[10] A. M. Nakhushev, Elementy drobnogo ischisleniya i ikh primenenie [Elements of Fractional Calculus and Their Application], Kabardino-Balkarsk. Nauchn. Tsentr Ross. Akad. Nauk, Nalchik, 2003, 299 pp. (In Russian)

[11] V. D. Beybalaev, “One-step methods on the solution of the Cauchy problem for ordinary differential equations with fractional order derivatives”, Vestnik Dagestanskogo gosudarstvennogo universiteta, 2011, no. 6, 67–72 (In Russian)

[12] U. G. Pirumov, Chislennye metody [Numerical methods], Drofa, Moscow, 2004, 224 pp. (In Russian)

[13] R. R. Nigmatullin, “Fractional integral and its physical interpretation”, Theoret. and Math. Phys., 90:3 (1992), 242–251 | DOI | MR | Zbl