Abstract
In this paper, we give an analytical solution of a fractional wave equation for a vibrating string with Caputo time fractional derivatives. We obtain the exact solution in terms of three parameter Mittag-Leffler function. Furthermore, some examples of the main result are exhibited.
Highlights
In recent years, fractional calculus has been one of the most popular topics in research [1,2,3,4,5]
Many authors have obtained the solutions of time fractional diffusionwave equations in a bounded domain in terms of the Mittag-Leffler type functions
3 Analytical results for the problem we investigate the analytical solution of the proposed problem (1)–(3)
Summary
Fractional calculus has been one of the most popular topics in research [1,2,3,4,5]. With Caputo time fractional derivatives C∗γ and C∗α of orders 1 < γ < 2 and 0 < α < 1, respectively, using the conditions w(x, t)|x=0 = h1(t), w(x, t)|x=l = h2(t). 2, definitions and properties of Mittag-Leffler functions and fractional integrals and derivatives are presented. The four parameter Mittag-Leffler function [49] was defined by. The Laplace transform of the Mittag-Leffler functions (6) is represented by (see [47, 50]). Note that the following expression gives us the relationship between the Caputo fractional derivative (10) and the Riemann–Liouville fractional integral operator (9) (see [4]). We give the Laplace transform for the Caputo fractional derivative in the following formula (see [5, 52]): n–1.
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