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 $, and introduced a QSC-$ 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 $ \mathcal{O}(\tau^{\min{\{3-\alpha^*-\alpha(0), \ 2\}}} + \Delta x^{2}+\Delta y^{2}) $, where $ \Delta x $, $ \Delta y $, and $ \tau $ denoted the space-time mesh sizes. $ \alpha^{*} $ 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.
Citation: Jun Liu, Yue Liu, Xiaoge Yu, Xiao Ye. An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition[J]. Electronic Research Archive, 2025, 33(2): 642-666. doi: 10.3934/era.2025030
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 $, and introduced a QSC-$ 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 $ \mathcal{O}(\tau^{\min{\{3-\alpha^*-\alpha(0), \ 2\}}} + \Delta x^{2}+\Delta y^{2}) $, where $ \Delta x $, $ \Delta y $, and $ \tau $ denoted the space-time mesh sizes. $ \alpha^{*} $ 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.
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