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

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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-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 $\mathcal{O}\left(\tau^{\min \left\{3-\alpha^{*}-\alpha(0), 2\right\}}+\Delta x^{2}+\Delta y^{2}\right)$, where ∆x, ∆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-L1⁺ 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.

Keywords

multi-term variable fractional order mobile-immobile equations / Neumann boundary condition / quadratic spline collocation / L1⁺ method / numerical analysis / fast computation

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