Mathematical analysis and symmetric fractional-order reduction method for diffusion-wave equations

In this work, our aim is to introduce a symmetric fractional-order reduction (SFOR) method to develop numerical algorithms on nonuniform temporal meshes for fractional wave equations under lower regularity assumptions. The $L$-type methods--including $L1$ and $L2$-$1_\sigma$ schemes--are specifically designed for diffusion-wave equations, and we propose novel optimal parameter selections tailored to nonuniform meshes. Finally, several numerical experiments are conducted to validate the efficiency and accuracy of the algorithms.