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, <middle dot> <middle dot> <middle dot>, P, and introduced a QSC-L1+ scheme by applying the quadratic spline collocation (QSC) method along the spatial direction and using the L1+formula for the temporal direction. This new scheme was shown to be unconditionally stable and convergent with the accuracy O(tau min {3-alpha & lowast;-alpha(0), 2} + triangle x2 + triangle y2), where triangle x, triangle y, and tau denoted the space-time mesh sizes. alpha & lowast; 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-L1+ 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.