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An Efficient QSC Approximation of Variable-Order Time-Fractional Mobile-Immobile Diffusion Equations with Variably Diffusive Coefficients

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Abstract

In this paper, we propose a QSC-L1 method to solve the two-dimensional variable-order time-fractional mobile-immobile diffusion (TF-MID) equations with variably diffusive coefficients, in which the quadratic spline collocation (QSC) method is employed for the spatial discretization, and the classical L1 formula is used for the temporal discretization. We show that the method is unconditionally stable and convergent with first-order in time and second-order in space with respect to some discrete and continuous \(L^2\) norms. Then, combined with the reduced basis technique, an efficient QSC-L1-RB method is proposed to improve the computational efficiency. Numerical examples are attached to verify the convergence orders, and also the method is applied to identify parameters of the variable-order TF-MID equations. Numerical results confirm the contributions of the reduced basis technique, even when the observation data is contaminated by some levels of random noise.

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Funding

The work of H. Fu was supported in part by the National Natural Science Foundation of China (Nos. 11971482, 12131014), the Fundamental Research Funds for the Central Universities (No. 202264006), and by the OUC Scientific Research Program for Young Talented Professionals. The work of J. Liu was supported in part by the Shandong Provincial Natural Science Foundation (Nos. ZR2021MA020, ZR2020MA039), the Fundamental Research Funds for the Central Universities (Nos. 22CX03016A, 20CX05011A), and the Major Scientific and Technological Projects of CNPC (No. ZD2019-184-001).

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  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China

    Jun Liu

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, Shandong, China

    Hongfei Fu

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  1. Jun Liu
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Correspondence to Hongfei Fu.

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Liu, J., Fu, H. An Efficient QSC Approximation of Variable-Order Time-Fractional Mobile-Immobile Diffusion Equations with Variably Diffusive Coefficients. J Sci Comput 93, 44 (2022). https://doi.org/10.1007/s10915-022-02007-2

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