Eur. Phys. J. Plus (2017) 132: 515
https://doi.org/10.1140/epjp/i2017-11796-9

Regular Article

Application of fractional derivative with exponential law to bi-fractional-order wave equation with frictional memory kernel

¹ Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, Mexico
² CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

^* e-mail: jgomez@cenidet.edu.mx

Received: 28 June 2017
Accepted: 14 November 2017
Published online: 11 December 2017

Abstract

Analytical solutions of the wave equation with bi-fractional-order and frictional memory kernel of Mittag-Leffler type are obtained via Caputo-Fabrizio fractional derivative in the Liouville-Caputo sense. Through the method of separation of variables and Laplace transform method we derive closed-form solutions and establish fundamental solutions. Special cases with homogeneous Dirichlet boundary conditions and nonhomogeneous initial conditions, as well as for the external force are considered. Numerical simulations of the special solutions were done and novel behaviors are obtained.