@article{Liu2025, 
author = {Jun Liu and Yue Liu and Xiaoge Yu and Xiao Ye},
title = {An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {2},
pages = {642-666},
keywords = {multi-term variable fractional order mobile-immobile equations, Neumann boundary condition, quadratic spline collocation, L1+ method, numerical analysis, fast computation},
url = {https://www.sciopen.com/article/10.3934/era.2025030},
doi = {10.3934/era.2025030},
abstract = {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        α          m        (  t  ) for    m  =  1  ,  2  ,  ⋯  ,  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        O    (      τ          min              {        3        −                  α          ∗                −        α        (        0        )        ,                2        }              +  Δ      x          2        +  Δ      y          2        ), where    Δ  x,    Δ  y, and    τ denoted the space-time mesh sizes.          α          ∗       was the maximum of        α          m        (  t  ) over the time interval, and    α  (  0  ) was the maximum of        α          m        (  0  ) in all values of    m. The QSC-   L      1    +   scheme, under certain appropriate conditions on        α          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.}
}