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 : Entrepreneurship and Africa’s Cultural Context , 2024 , 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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