Authors: Atanacković, Teodor
Pilipović, Stevan
Zorica, Dušan 
Title: Time distributed-order diffusion-wave equation. I. Volterra-type equation
Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume: 465
Issue: 2106
First page: 1869
Last page: 1891
Issue Date: 8-Jun-2009
Rank: M21
ISSN: 1364-5021
DOI: 10.1098/rspa.2008.0445
A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributedorder fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.
Keywords: Diffusion-wave equation | Distributed-order fractional derivative | Fractional derivative | Volterra equation
Publisher: The Royal Society
Project: Serbian Ministry of Sciences, Grants 144019A and 144016

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