0
Research Papers

Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation

[+] Author and Article Information
Yuri Luchko

Professor of Mathematics
Department of Mathematics,
Physics, and Chemistry,
Beuth Technical University of Applied Sciences,
Berlin 13353, Germany
e-mail: luchko@beuth-hochschule.de

Francesco Mainardi

Professor of Mathematical Physics,
Department of Physics and Astronomy,
Bologna University, and INFN,
Bologna 40126, Italy
e-mails: francesco.mainardi@unibo.it; francesco.mainardi@bo.infn.it

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 28, 2013; final manuscript received February 12, 2014; published online July 25, 2014. Assoc. Editor: Thomas J. Royston.

J. Vib. Acoust 136(5), 050904 (Jul 25, 2014) (7 pages) Paper No: VIB-13-1301; doi: 10.1115/1.4026892 History: Received August 28, 2013; Revised February 12, 2014

Abstract

In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order $β,1≤β≤2$ are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity 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 in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional diffusion-wave equation is nonrelativistic like the classical diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other.

<>

References

Gorenflo, R., and Mainardi, F., 1997, “Fractional Calculus, Integral and Differential Equations of Fractional Order,” Fractals and Fractional Calculus in Continuum Mechanics, A.Carpinteri and F.Mainardi, eds., Springer-Verlag, Wien, Germany, pp. 223–276.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Nigmatullin, R. R., 1986, “The Realization of the Generalized Transfer Equation in a Medium With Fractal Geometry,” Phys. Stat. Sol. B, 133(1), pp. 425–430.
Mainardi, F.1995, “Fractional Diffusive Waves in Viscoelastic Solids,” IUTAM Symposium—Nonlinear Waves in Solids, J. L.Wegner and F. R.Norwood, eds., ASME/AMR, Fairfield, NJ, pp. 93–97.
Pipkin, A. C., 1986, Lectures on Viscoelastic Theory, Springer-Verlag, New York.
Kreis, A., and Pipkin, A. C., 1986, “Viscoelastic Pulse Propagation and Stable Probability Distributions,” Quart. Appl. Math., 44, pp. 353–360.
Mainardi, F., 1994, “On the Initial Value Problem for the Fractional Diffusion-Wave Equation,” Waves and Stability in Continuous Media, S.Rionero and T.Ruggeri, eds., World Scientific, Singapore, pp. 246–251.
Mainardi, F., 1995, “The Time Fractional Diffusion-Wave Equation,” Radiofisika, 38, pp. 20–36.
Mainardi, F., 1996, “Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena,” Chaos Solitons Fract., 7(9), pp. 1461–1477.
Mainardi, F., 1996, “The Fundamental Solutions for the Fractional Diffusion-Wave Equation,” Appl. Math. Lett., 9(6), pp. 23–28.
Mainardi, F., 1997, “Fractional Calculus, Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A.Carpinteri and F.Mainardi, eds., Springer-Verlag, Wien, Germany, pp. 291–348.
Wyss, W., 1986, “Fractional Diffusion Equation,” J. Math. Phys., 27(11), pp. 2782–2785.
Schneider, W. R., and Wyss, W., 1989, “Fractional Diffusion and Wave Equations,” J. Math. Phys., 30(1), pp. 134–144.
Fujita, Y., 1990, “Integrodifferential Equation Which Interpolates the Heat Equation and the Wave Equation, Part I,” Osaka J. Math., 27(2), pp. 309–321. Available at: http://projecteuclid.org/euclid.ojm/1200782311.
Fujita, Y., 1990, “Integrodifferential Equation Which Interpolates the Heat Equation and the Wave Equation, Part II,” Osaka J. Math., 27, pp. 797–804. Available at: http://projecteuclid.org/euclid.ojm/1200782677.
Prüss, J., 1993, Evolutionary Integral Equations and Applications, Birkhäuser-Verlag, Basel, Switzerland.
Mainardi, F., Luchko, Y., and Pagnini, G., 2001, “The Fundamental Solution of the Space-Time Fractional Diffusion Equation,” Fract. Calc. Appl. Anal., 4(2), pp. 153–192.
Mainardi, F., Pagnini, G., and Saxena, R. K., 2005, “Fox H Functions in Fractional Diffusion,” J. Comp. Appl. Math., 178(1-2), pp. 321–331.
Luchko, Y., 2009, “Boundary Value Problems for the Generalized Time-Fractional Diffusion Equation of Distributed Order,” Fract. Calc. Appl. Anal., 12(4), pp. 409–422.
Luchko, Y., 2010, “Some Uniqueness and Existence Results for the Initial-Boundary-Value Problems for the Generalized Time-Fractional Diffusion Equation,” Comput. Math. Appl., 59(5), pp. 1766–1772.
Luchko, Y., 2011, “Initial-Boundary-Value Problems for the Generalized Multi-Term Time-Fractional Diffusion Equation,” J. Math. Anal. Appl., 374(2), pp. 538–548.
Luchko, Y., 2012, “Initial-Boundary-Value Problems for the One-Dimensional Time-Fractional Diffusion Equation,” Fract. Calc. Appl. Anal., 15(1), pp. 141–160.
Luchko, Y., 2013, “Fractional Wave Equation and Damped Waves,” J. Math. Phys., 54(3), p. 031505.
Luchko, Y., and Punzi, A., 2011, “Modeling Anomalous Heat Transport in Geothermal Reservoirs Via Fractional Diffusion Equations,” Int. J. Geomath., 1(2), pp. 257–276.
Buckwar, E., and Luchko, Y., 1998, “Invariance of a Partial Differential Equation of Fractional Order Under the Lie Group of Scaling Transformations,” J. Math. Anal. Appl., 227(1), pp. 81–97.
Engler, H., 1997, “Similarity Solutions for a Class of Hyperbolic Integrodifferential Equations,” Diff. Integral Eq., 10(5), pp. 815–840. Available at: http://projecteuclid.org/euclid.die/1367438621.
Fujita, Y., 1990, “Cauchy Problems of Fractional Order and Stable Processes,” Japan J. Appl. Math., 7(3), pp. 459–476.
Gorenflo, R., Luchko, Y., and Mainardi, F., 1999, “Analytical Properties and Applications of the Wright Function,” Fract. Calc. Appl. Anal., 2(4), pp. 383–414.
Gorenflo, R., Luchko, Y., and Mainardi, F., 2000, “Wright Functions as Scale-Invariant Solutions of the Diffusion-Wave Equation,” J. Comput. Appl. Math., 118(1-2), pp. 175–191.
Luchko, Y., and Gorenflo, R., 1998, “Scale-Invariant Solutions of a Partial Differential Equation of Fractional Order,” Fract. Calc. Appl. Anal., 3(1), pp. 63–78.
Mainardi, F., and Tomirotti, M., 1997, “Seismic Pulse Propagation With Constant Q and Stable Probability Distributions,” Annali di Geofisica, 40(5), pp. 1311–1328.
Luchko, Y., Mainardi, F., and Povstenko, Y., 2013, “Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation,” Comput. Math. Appl., 66(5), pp. 774–784.
Luchko, Y., and Mainardi, F., 2013, “Some Properties of the Fundamental Solution to the Signalling Problem for the Fractional Diffusion-Wave Equation,” Cent. Eur. J. Phys.11(6), pp. 666–675.
Luchko, Y., 2009, “Maximum Principle for the Generalized Time-Fractional Diffusion Equation,” J. Math. Anal. Appl., 351(1), pp. 218–223.
Luchko, Y., 2011, “Maximum Principle and Its Application for the Time-Fractional Diffusion Equations,” Fract. Calc. Appl. Anal., 14(1), pp. 110–124.
Gorenflo, R., Loutchko, J., and Luchko, Y., 2002, “Computation of the Mittag–Leffler Function and Its Derivatives,” Fract. Calc. Appl. Anal., 5(4), pp. 491–518.
Luchko, Y., 2008, “Algorithms for Evaluation of the Wright Function for the Real Arguments' Values,” Fract. Calc. Appl. Anal., 11(1), pp. 57–75.
Mainardi, F., 2010, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London.

Figures

Fig. 1

The Green function Gc(x;ν):=Gc(x,1;ν). Plots for several different values of ν.

Fig. 2

The Green function Gs(x;ν):=Gs(x,1;ν). Plots for several different values of ν.

Fig. 3

Propagation velocity Vc(t,ν) of the maximum point of Gc in the log-lin scale

Fig. 4

Maximum locations and maximum values of Gc(x,t;ν)

Fig. 5

Maximum locations and maximum values of the Green function Gc(x,1;ν)

Fig. 6

Product of the maximum locations and maximum values of the Green function Gc(x,1;ν)

Fig. 7

Propagation velocity Vs(t,ν) of the maximum point of Gs

Fig. 8

Maximum locations and maximum values of the Green function Gs(x,1;ν)

Fig. 9

Location of the center of gravity of the Green function Gc(|x|,1;ν)

Fig. 10

Propagation velocity Vcg(t,ν) of the center of gravity of Gc

Fig. 11

Location of the center of gravity of the Green function Gs(x,1;ν)

Fig. 12

Propagation velocity Vsg(t,ν) of the center of gravity of Gs

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections