Time-fractional variable-order telegraph equation involving operators with Mittag-Leffler kernel

ABSTRACT

In this paper, we have generalized the time-fractional telegraph equation involving operators with Mittag-Leffler kernel of variable order in Liouville-Caputo sense. The fractional variable-order equation was solved numerically via Crank-Nicholson scheme. We present the existence and uniqueness of the solution. Numerical simulations of the special solutions were done and new behaviors are obtained.

KEYWORDS:

• Fractional telegraph equation
• Mittag-Leffler function
• Crank-Nicholson scheme
• Fractional variable-order derivative

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 26A33
• 65R10
• 70Hxx

Acknowledgments

The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the paper.

Disclosure statement

The authors declare that there is no conflict of interest regarding the publication of this paper.

Additional information

Funding

José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

Notes on contributors

J. F. Gómez-Aguilar

J. F. Gómez-Aguilar received the BS and MEng degrees in electrical engineering from Guanajuato University, Guanajuato, Mexico, in 2005 and 2007, respectively, and the PhD degree in physics from División de Ciencias e Ingenierías, Guanajuato University, in 2012. He is currently a Full Research Professor with the Electronics Engineering Department commissioned for the CONACyT in the Centro Nacional de Investigación y Desarrollo Tecnologico, Tecnológico Nacional de México, Cuernavaca Morelos, México. His current research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbations methods, image and signal processing, control, and power quality analysis.

Abdon Atangana

Abdon Atangana obtained his BS degree in pure mathematics and his Honors degree and MS degree in Applied Mathematics with distinction from the University of Free State, South Africa. He also obtained his PhD degree in applied mathematics from the Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa. He is serving as editor of some international journals, and also reviewer of more than 100 international accredited journals. He has been the Lead and Guest Editor of some special issues, and has presented and participated in more than 15 international conferences. His research interests are methods and applications of partial and ordinary differential equations, fractional differential equations, perturbations methods, asymptotic methods, iterative methods, and groundwater modeling. He is the author of the book “‘Derivative with new parameter: theory, methods and applications”, published by Academic Press Elsevier.