PL EN


Preferences help
enabled [disable] Abstract
Number of results
Journal
2013 | 11 | 6 | 666-675
Article title

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

Content
Title variants
Languages of publication
EN
Abstracts
EN
In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.
Publisher

Journal
Year
Volume
11
Issue
6
Pages
666-675
Physical description
Dates
published
1 - 6 - 2013
online
9 - 10 - 2013
Contributors
author
  • Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, D-13353, Berlin, Germany, luchko@beuth-hochschule.de
References
  • [1] E. Buckwar, Yu Luchko, J. Math. Anal. Appl. 227, 81 (1998) http://dx.doi.org/10.1006/jmaa.1998.6078[Crossref]
  • [2] Y. Fujita, Osaka J. Math. 27, 309, 797 (1990)
  • [3] Y. Fujita, Japan J. Appl. Math. 7, 459 (1990) http://dx.doi.org/10.1007/BF03167854[Crossref]
  • [4] R. Gorenflo, R. Rutman, On ultraslow and intermediate processes, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and Special Functions, Sofia 1994 (Science Culture Technology Publ., Singapore, 1995) 61
  • [5] R. Gorenflo, Yu. Luchko, F. Mainardi, Fract. Calc. Appl. Anal. 2, 383 (1999).
  • [6] R. Gorenflo, Yu. Luchko, F. Mainardi, J. Comput. Appl. Math. 11, 175 (2000) http://dx.doi.org/10.1016/S0377-0427(00)00288-0[Crossref]
  • [7] R. Gorenflo, J. Loutchko, Yu. Luchko, Fract. Calc. Appl. Anal. 5, 491 (2002)
  • [8] A.A. Kilbas, M. Saigo, H-transforms. Theory and Applications (Chapman and Hall/CRC, Boca Raton, FL, 2004) http://dx.doi.org/10.1201/9780203487372[Crossref]
  • [9] A.N. Kochubei, Diff. Equat. 25, 967 (1989) [English translation from the Russian Journal Differentsial’nye Uravneniya]
  • [10] A.N. Kochubei, Diff. Equat. 26, 485 (1990) [English translation from the Russian Journal Differentsial’nye Uravneniya]
  • [11] A. Kreis, A.C. Pipkin, Quart. Appl. Math. 44, 353 (1986)
  • [12] Yu. Luchko, R. Gorenflo, Fract. Calc. Appl. Anal. 1, 63 (1998)
  • [13] Yu. Luchko, Fract. Calc. Appl. Anal. 11, 57 (2008)
  • [14] Yu. Luchko, J. Math. Anal. Appl. 351, 218 (2009) http://dx.doi.org/10.1016/j.jmaa.2008.10.018[Crossref]
  • [15] Yu. Luchko, Computers and Mathematics with Applications 59, 1766 (2010) http://dx.doi.org/10.1016/j.camwa.2009.08.015[Crossref]
  • [16] Yu. Luchko, J. Math. Anal. Appl. 374, 538 (2011) http://dx.doi.org/10.1016/j.jmaa.2010.08.048[Crossref]
  • [17] Yu. Luchko, Fract. Calc. Appl. Anal. 14, 110 (2011)
  • [18] Yu. Luchko, Fract. Calc. Appl. Anal. 15, 141 (2012)
  • [19] Yu. Luchko, J. Math. Phys. 54, 031505/1 (2013)
  • [20] Yu. Luchko, F. Mainardi, Yu. Povstenko, Comput. Math. Appl., in press (2013) DOI: 10.1016/j.camwa.2013.01.005 [Crossref]
  • [21] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in: S. Rionero and T. Ruggeri (Eds.), Waves and Stability in Continuous Media (World Scientific, Singapore, 1994) 246
  • [22] F. Mainardi, Chaos Solitons Fractals 7, 1461 (1996) http://dx.doi.org/10.1016/0960-0779(95)00125-5[Crossref]
  • [23] F. Mainardi, Appl. Math. Lett. 9, 23 (1996) http://dx.doi.org/10.1016/0893-9659(96)00089-4[Crossref]
  • [24] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. (Imperial College Press, London, 2010) http://dx.doi.org/10.1142/9781848163300[Crossref]
  • [25] F. Mainardi, Forum der Berliner Mathematischer Gesellschaft 19, 20 (2011)
  • [26] F. Mainardi, M. Tomirotti, On a special function arising in the time fractional diffusion-wave equation, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and Special Functions, Sofia 1994 (Science Culture Technology, Singapore, 1995) 171
  • [27] F. Mainardi, M. Tomirotti, Ann. Geofis. 40, 1311 (1997)
  • [28] F. Mainardi, Yu. Luchko, G. Pagnini, Fract. Calc. Appl. Anal. 4, 153 (2001)
  • [29] F. Mainardi, G. Pagnini, R.K. Saxena J. Comp. Appl. Math. 178, 321 (2005) http://dx.doi.org/10.1016/j.cam.2004.08.006[Crossref]
  • [30] A.M. Mathai, H.J. Haubold, Special Functions for Applied Scientists (Springer Verlag, New York, 2008) http://dx.doi.org/10.1007/978-0-387-75894-7[Crossref]
  • [31] A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-function: Theory and Applications (Springer Verlag, New York, 2010) http://dx.doi.org/10.1007/978-1-4419-0916-9[Crossref]
  • [32] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
  • [33] J. Prüss, Evolutionary Integral Equations and Applications (Birkhauser Verlag, Basel, 1993) http://dx.doi.org/10.1007/978-3-0348-8570-6[Crossref]
  • [34] R.L. Schilling, R. Song, Z. Vondracek, Bernstein Functions. Theory and Applications, (De Gruyter, Berlin, 2010)
  • [35] W.R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989) http://dx.doi.org/10.1063/1.528578[Crossref]
  • [36] W. Wyss, J. Math. Phys. 27, 2782 (1986) http://dx.doi.org/10.1063/1.527251[Crossref]
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_s11534-013-0247-8
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.