An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition

In this work, we aimed at a kind of multi-term variable-order time fractional mobile-immobile diffusion (TF-MID) equation satisfying the Neumann boundary condition, with fractional orders ${\alpha }^m(t)$ for $m=1,2,\cdots ,P$, and introduced a QSC- $L{1}^{+}$ scheme by applying the quadratic spline collocation (QSC) method along the spatial direction and using the $L{1}^{+}$ formula for the temporal direction. This new scheme was shown to be unconditionally stable and convergent with the accuracy $\mathcal{O}({\tau }^{min\{3-{\alpha }^{\ast }-\alpha (0),2\}}+\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2)$, where $\mathrm{\Delta }x$, $\mathrm{\Delta }y$, and $\tau$ denoted the space-time mesh sizes. ${\alpha }^{\ast }$ was the maximum of ${\alpha }^m(t)$ over the time interval, and $\alpha (0)$ was the maximum of ${\alpha }^m(0)$ in all values of $m$. The QSC- $L{1}^{+}$ scheme, under certain appropriate conditions on ${\alpha }^m(t)$, is capable of attaining a second order convergence in time, even on a uniform space-time grid. Additionally, we also implemented a fast computation approach which leveraged the exponential-sum-approximation technique to increase the computational efficiency. A numerical example with different fractional orders was attached to confirm the theoretical findings.