Numerical approximation for variable-exponent fractional diffusion-wave equation

This work presents two numerical schemes for the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. The main difficulty we overcome lies in that the variable-exponent Abel kernel may not be positive definite or monotonic, and the stability and error estimate of both schemes are proved, with $\alpha(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings.