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

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.