Distributed-order wave equations with composite time fractional derivative. (English) Zbl 1513.35534
Summary: In this paper we investigate the solution of generalized distributed-order wave equations with composite time fractional derivative and external force, by using the Fourier-Laplace transform method. We represent the corresponding solutions in terms of infinite series in three parameter (Prabhakar), Mittag-Leffler and Fox \(H\)-functions, as well as in terms of the so-called Prabhakar integral operator. Generalized uniformly distributed-order wave equation is analysed by using the Tauberian theorem, and the mean square displacement is graphically represented by applying a numerical Laplace inversion algorithm. The numerical results and asymptotic behaviors are in good agreement. Some interesting examples of distributed-order wave equations with special external forces by using the Dirac delta function are also considered.
MSC:
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 33E12 | Mittag-Leffler functions and generalizations |
| 40E05 | Tauberian theorems |
Keywords:
distributed-order equations; composite derivative; wave equation; Mittag-Leffler functions; fox \(H\)-function; numerical Laplace inversion; Tauberian theoremReferences:
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