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Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation

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Abstract

In this paper, we consider a non-homogeneous time–space-fractional telegraph equation in n-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm–Liouville operator defined in terms of right and left fractional Riemann–Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.

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Acknowledgements

The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within projects UIDB/04106/2020 and UIDP/04106/2020. N. Vieira was also supported by FCT via the 2018 FCT program of Stimulus of Scientific Employment - Individual Support (Ref: CEECIND/01131/2018).

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Authors and Affiliations

  1. School of Technology and Management, Polytechnic of Leiria, 2411-901, Leiria, Portugal

    M. Ferreira

  2. CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    M. Ferreira, M. M. Rodrigues & N. Vieira

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  1. M. Ferreira
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  2. M. M. Rodrigues
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  3. N. Vieira
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Correspondence to N. Vieira.

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Communicated by Daniel Aron Alpay.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.

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Ferreira, M., Rodrigues, M.M. & Vieira, N. Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation. Complex Anal. Oper. Theory 15, 87 (2021). https://doi.org/10.1007/s11785-021-01125-3

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