Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T18:25:03.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Behaviour of the Time-Fractional Telegraph Equation

Part of: Stochastic processes Qualitative behavior

Published online by Cambridge University Press:  30 January 2018

Vicente Vergara
Vicente Vergara*
Affiliation:
Universidad de Tarapacá
*
∗ Postal address: Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta N. 1520, Arica, Chile. Email address: vvergaraa@uta.cl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain the long-time behaviour to the variance of the distribution process associated with the solution of the telegraph equation. To this end, we use a version of the Karamata-Feller Tauberian theorem.

Keywords

Telegraph equationfractional derivativeMittag-Leffler functionTauberian theorem

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 45K05: Integro-partial differential equations 45M05: Asymptotics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 890 - 893
Copyright
© Applied Probability Trust 

References

Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gripenberg, G., Londen, S.-O. and Staffans, O. (1990). Volterra Integral and Functional Equations (Encyclopedia Math. Appl. 34). Cambridge University Press.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations (North-Holland Math. Studies 204). Elsevier, Amsterdam.Google Scholar
Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. {Prob. Theory Relat. Fields} 128, 141–160.Google Scholar
Podlubny, I. (1999). Fractional Differential Equations (Math. Sci. Eng. 198). Academic Press, San Diego, CA.Google Scholar
Prüss, J. (1993). Evolutionary Integral Equations and Applications. Birkhäuser, Basel.Google Scholar
Samko, S.-G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, Yverdon.Google Scholar
Vergara, V. and Zacher, R. (2013). Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods. Submitted.Google Scholar