Existence and uniqueness of solution to a class of nonlinear variable fractional-order wave equations

Abstract

In this work, we deal with the results on existence and uniqueness for a class of fractional wave equations involving nonlinear variable fractional-order derivative by the Rothe-Galerkin’s method. Firstly, we give the weak form of the wave equation without impulsive points and use the Grönwall’s lemma to demonstrate the uniqueness of the solution. Next, we introduce an approximating problem based on a time-discrete scheme in the backward sense, and predict a priori estimates for the time-discrete solution that converges to the solution of the original equation. Finally, we investigate the existence of weak solutions for the nonlinear variable fractional-order impulsive wave equation.