On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations

In this contribution, a wave equation with time-dependent variable-order fractional damping term and nonlinear source is considered. Avoiding the circumstances of expressing equations via closed-form expressions in terms special functions, we investigate existence uniqueness problem Rothe’s method. First, weak formulation for considered proposed. Then, solution established by employing Grönwall’s lemma. The Rothe scheme’s basic idea to use functions extend solutions on single-time steps over entire time frame. Inspired that, next introduce uniform mesh time-discrete scheme based discrete convolution approximation backward sense. By applying some reasonable assumptions given data, can predict priori estimates solution. Employing these side leads proof solution’s whole interval. Finally, full discretisation introduced invoking Galerkin spectral techniques spatial direction, numerical examples are given.