Research Article

Space-dependent variable-order time-fractional wave equation: Existence and uniqueness of its weak solution

Published in: Quaestiones Mathematicae
Volume 46 , issue 8 , 2023 , pages: 1695–1715

DOI: 10.2989/16073606.2022.2110959

Author(s): K. Van Bockstal Ghent University, Belgium , A.S. Hendy Ural Federal University, Russia , M.A. Zaky National Research Centre, Egypt

Keywords: 35A15 , 35R11 , 47G20 , 65M12 , Time fractional wave equation , variable coeﬃcients , uniqueness and existence , non-autonomous , variable-order , Rothe’s time discretization

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Abstract

The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coeﬃcients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional diﬀerential operator originates in the modelling of viscoelastic materials. The global in time existence of a unique weak solution to the model problem has been proved under appropriate conditions on the data. Rothe’s time discretization method is applied to achieve that purpose.

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