Journal
COMPUTERS & MATHEMATICS WITH APPLICATIONS
Volume 62, Issue 3, Pages 1554-1561Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.camwa.2011.04.042
Keywords
Wave equation; Frictional power-law memory kernel; Caputo time fractional derivative; Mittag-Leffler function; Fractional integral operator; Fractional differential operator
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In this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power-law memory kernel. The exact solution is obtained in terms of the Mittag-Leffler type functions and a generalized integral operator containing a four parameter Mittag-Leffler function in the kernel. The method of separation of the variables, Laplace transform method and Sturm-Liouville problem are used to solve the equation analytically. The asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the Tauberian theorems. The proposed model may be used for describing processes where the memory effects of complex media could not be neglected. (C) 2011 Elsevier Ltd. All rights reserved.
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