Symmetric fractional order reduction method with $L1$ scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in time. We derive the $\alpha$-robust $\mathit{\text{a}}\mathit{\text{priori}}$ bound and $\mathit{\text{a}}\mathit{\text{priori}}$ error estimate for the fully-discrete solution in $L^{\mathrm{\infty }}(H_0^1(\mathrm{\Omega }))$ norm, where $\alpha \in (1,2)$ is the order of time fractional derivative. Finally, we perform some numerical experiments to verify the theoretical results.