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The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coecients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional dierential 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.

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