Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivatives

Nelson Vieira; M. Manuela Rodrigues; Milton Ferreira; Nelson Vieira; M. Manuela Rodrigues; Milton Ferreira

doi:10.3934/era.2022184

Electronic Research Archive

2022, Volume 30, Issue 10: 3595-3631. doi: 10.3934/era.2022184

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Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivatives

1.
CIDMA-Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, Aveiro 3810-193, Portugal
2.
School of Technology and Management, Polytechnic of Leiria, Campus 2-Morro do Lena, Alto do Vieiro, Leiria P-2411-901, Portugal

Academic Editor: Arran Fernandez

Received: 30 December 2021 Revised: 15 July 2022 Accepted: 24 July 2022 Published: 29 July 2022

In this paper, we consider the time-fractional telegraph equation of distributed order in higher spatial dimensions, where the time derivative is in the sense of Hilfer, thus interpolating between the Riemann-Liouville and the Caputo fractional derivatives. By employing the techniques of the Fourier, Laplace, and Mellin transforms, we obtain a representation of the solution of the Cauchy problem associated with the equation in terms of convolutions involving functions that are Laplace integrals of Fox H-functions. Fractional moments of the first fundamental solution are computed and for the special case of double-order distributed it is analyzed in detail the asymptotic behavior of the second-order moment, by application of the Tauberian Theorem. Finally, we exhibit plots of the variance showing its behavior for short and long times, and for different choices of the parameters along small dimensions.
- time-fractional telegraph equation,
- distributed order,
- Hilfer fractional derivative,
- integral transforms,
- fox H-function,
- fractional moments,
- Tauberian Theorem
Citation: Nelson Vieira, M. Manuela Rodrigues, Milton Ferreira. Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivatives[J]. Electronic Research Archive, 2022, 30(10): 3595-3631. doi: 10.3934/era.2022184

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Abstract

In this paper, we consider the time-fractional telegraph equation of distributed order in higher spatial dimensions, where the time derivative is in the sense of Hilfer, thus interpolating between the Riemann-Liouville and the Caputo fractional derivatives. By employing the techniques of the Fourier, Laplace, and Mellin transforms, we obtain a representation of the solution of the Cauchy problem associated with the equation in terms of convolutions involving functions that are Laplace integrals of Fox H-functions. Fractional moments of the first fundamental solution are computed and for the special case of double-order distributed it is analyzed in detail the asymptotic behavior of the second-order moment, by application of the Tauberian Theorem. Finally, we exhibit plots of the variance showing its behavior for short and long times, and for different choices of the parameters along small dimensions.

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