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.
| [1] |
B. B. Mandelbrot, The fractal geometry of nature, Am. J. Phys., 51 (1983), 286–287. https://doi.org/10.1119/1.13295 doi: 10.1119/1.13295
|
| [2] |
S. L. Lu, F. J. Molz, G. J. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water Resour. Res., 38 (2002), 1165. https://doi.org/10.1029/2001wr000624 doi: 10.1029/2001wr000624
|
| [3] |
S. G. Samko, B. Ross, Integration and differentiation to a variable fractional-order, Integral Transform Spec. Funct., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
|
| [4] |
Z. Li, H. Wang, R. Xiao, S. Yang, A variable-order fractional differential equation model of shape memory polymers, Chaos, Solitons Fractals, 102 (2017), 473–485. https://doi.org/10.1016/j.chaos.2017.04.042 doi: 10.1016/j.chaos.2017.04.042
|
| [5] |
H. G. Sun, W. Chen, Y. Q. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), 4586–4592. https://doi.org/10.1016/j.physa.2009.07.024 doi: 10.1016/j.physa.2009.07.024
|
| [6] |
H. Sheng, H. G. Sun, C. Coopmans, Y. Q. Chen, G. W. Bohannan, A physical experimental study of variable-order fractional integrator and differentiator, Eur. Phys. J. Spec. Top., 193 (2011), 93–104. https://doi.org/10.1140/epjst/e2011-01384-4 doi: 10.1140/epjst/e2011-01384-4
|
| [7] |
H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, C. F. M. Coimbra, Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere, J. Vib. Control, 14 (2008), 1659–1672. https://doi.org/10.1177/1077546307087397 doi: 10.1177/1077546307087397
|
| [8] |
H. Wang, X. C. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475 (2019), 1778–1802. https://doi.org/10.1016/j.jmaa.2019.03.052 doi: 10.1016/j.jmaa.2019.03.052
|
| [9] |
X. C. Zheng, H. Wang, Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains, Inverse Probl. Sci. Eng., 29 (2021), 1401–1411. https://doi.org/10.1080/17415977.2020.1849182 doi: 10.1080/17415977.2020.1849182
|
| [10] |
W. P. Bu, A. G. Xiao, W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422–441. https://doi.org/10.1007/s10915-017-0360-8 doi: 10.1007/s10915-017-0360-8
|
| [11] |
J. Guo, D. Xu, W. L. Qiu, A finite difference scheme for the nonlinear time-fractional partial integro-differential equation, Math. Methods Appl. Sci., 43 (2020), 3392–3412. https://doi.org/10.1002/mma.6128 doi: 10.1002/mma.6128
|
| [12] |
C. P. Xie, S. M. Fang, Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions, Math. Methods Appl. Sci., 43 (2020), 3473–3487. https://doi.org/10.1002/mma.6132 doi: 10.1002/mma.6132
|
| [13] |
S. J. Cheng, N. Du, H. Wang, Z. W. Yang, Finite element approximations to Caputo-Hadamard time-fractional diffusion equation with application in parameter identification, Fractal Fract., 6 (2022), 525. https://doi.org/10.3390/fractalfract6090525 doi: 10.3390/fractalfract6090525
|
| [14] |
W. Y. Kang, B. A. Egwu, Y. B. Yan, A. K. Pani, Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noise, IMA J. Numer. Anal., 42 (2022), 2301–2335. https://doi.org/10.1093/imanum/drab035 doi: 10.1093/imanum/drab035
|
| [15] |
Z. G. Zhao, Y. Y. Zheng, P. Guo, A Galerkin finite element method for a class of time-space fractional differential equation with nonsmooth data, J. Sci. Comput., 70 (2017), 386–406. https://doi.org/10.1007/s10915-015-0107-3 doi: 10.1007/s10915-015-0107-3
|
| [16] |
B. Bialecki, G. Fairweather, A. Karageorghis, J. Maack, A quadratic spline collocation method for the Dirichlet biharmonic problem, Numer. Algorithms, 83 (2020), 165–199. https://doi.org/10.1007/s11075-019-00676-z doi: 10.1007/s11075-019-00676-z
|
| [17] |
C. C. Christara, Quadratic spline collocation methods for elliptic partial differential equations, BIT Numer. Math., 34 (1994), 33–61. https://doi.org/10.1007/bf01935015 doi: 10.1007/bf01935015
|
| [18] |
J. Liu, C. Zhu, Y. P. Chen, H. F. Fu, A Crank-Nicolson ADI quadratic spline collocation method for two-dimensional Riemann-Liouville space-fractional diffusion equations, Appl. Numer. Math., 160 (2021), 331–348. https://doi.org/10.1016/j.apnum.2020.10.015 doi: 10.1016/j.apnum.2020.10.015
|
| [19] |
J. Liu, H. F. Fu, X. C. Chai, Y. N. Sun, H. Guo, Stability and convergence analysis of the quadratic spline collocation method for time-dependent fractional diffusion equations, Appl. Math. Comput., 346 (2019), 633–648. https://doi.org/10.1016/j.amc.2018.10.046 doi: 10.1016/j.amc.2018.10.046
|
| [20] |
Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
|
| [21] |
Z. Z. Sun, X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003 doi: 10.1016/j.apnum.2005.03.003
|
| [22] |
A. A. Alikhanov, A new difference scheme for the time fractional diffusion euqation, J. Comput. Phys., 280 (2015), 424–438. https://doi.org/10.1016/j.jcp.2014.09.031 doi: 10.1016/j.jcp.2014.09.031
|
| [23] |
M. F. She, D. F. Li, H. W. Sun, A transformed $L1$ method for solving the multi-term time-fractional diffusion problem, Math. Comput. Simul., 193 (2022), 584–606. https://doi.org/10.1016/j.matcom.2021.11.005 doi: 10.1016/j.matcom.2021.11.005
|
| [24] |
W. Q. Yuan, D. F. Li, C. J. Zhang, Linearized transformed $L1$ Galerkin FEMs with unconditional convergence for nonlinear time fractional Schr$\ddot{\rm{o}}$dinger equations, Numer. Math. Theory Methods Appl., 16 (2023), 348–369. https://doi.org/10.4208/nmtma.oa-2022-0087 doi: 10.4208/nmtma.oa-2022-0087
|
| [25] |
H. Chen, Y. Shi, J. W. Zhang, Y. M. Zhao, Sharp error estimate of a Gr$\ddot{\rm{u}}$nwald-Letnikov scheme for reaction-subdiffusion equations, Numer. Algoritms, 89 (2022), 1465–1477. https://doi.org/10.1007/s11075-021-01161-2 doi: 10.1007/s11075-021-01161-2
|
| [26] |
B. Y. Zhou, X. L. Chen, D. F. Li, Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85 (2020), 39. https://doi.org/10.1007/s10915-020-01350-6 doi: 10.1007/s10915-020-01350-6
|
| [27] |
J. C. Ren, H. L. Liao, J. W. Zhang, Z. M. Zhang, Sharp $H1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems, J. Comput. Appl. Math., 389 (2021), 113352. https://doi.org/10.1016/j.cam.2020.113352 doi: 10.1016/j.cam.2020.113352
|
| [28] |
X. C. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522–1545. https://doi.org/10.1093/imanum/draa013 doi: 10.1093/imanum/draa013
|
| [29] |
R. L. Du, Z. Z. Sun, H. Wang, Temporal second-order finite difference schemes for variable-order time-fractional wave equations, SIAM J. Numer. Anal., 60 (2022), 104–132. https://doi.org/10.1137/19m1301230 doi: 10.1137/19m1301230
|
| [30] |
H. M. Zhang, F. W. Liu, P. Zhuang, I. Turner, V. Anh, Numerical analysis of a new space-time variable fractional order advection-dispersion equation, Appl. Math. Comput., 242 (2014), 541–550. https://doi.org/10.1016/j.amc.2014.06.003 doi: 10.1016/j.amc.2014.06.003
|
| [31] |
H. Liu, X. C. Zheng, H. F. Fu, Analysis of a multi-term variable-order time-fractional diffusion equation and its Galerkin finite element approximation, J. Comput. Math., 40 (2022), 814–834. https://doi.org/10.4208/jcm.2102-m2020-0211 doi: 10.4208/jcm.2102-m2020-0211
|
| [32] |
J. Liu, H. F. Fu, An efficient QSC approximation of variable-order time-fractional mobile-immobile diffusion equations with variably diffusive coefficients, J. Sci. Comput., 93 (2022), 44. https://doi.org/10.1007/s10915-022-02007-2 doi: 10.1007/s10915-022-02007-2
|
| [33] |
B. Q. Ji, H. L. Liao, Y. Z. Gong, L. M. Zhang, Adaptive second-order Crank-Nicolson time stepping schemes for time-fractional molecular beam epitaxial growth models, SIAM J. Sci. Comput., 42 (2020), B738–B760. https://doi.org/10.1137/19m1259675 doi: 10.1137/19m1259675
|
| [34] |
S. D. Jiang, J. W. Zhang, Q. Zhang, Z. M. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.oa-2016-0136 doi: 10.4208/cicp.oa-2016-0136
|
| [35] |
J. L. Zhang, Z. W. Fang, H. W. Sun, Exponential-sum-approximation technique for variable-order time-fractional diffusion equations, J. Appl. Math. Comput., 68 (2022), 323–347. https://doi.org/10.1007/s12190-021-01528-7 doi: 10.1007/s12190-021-01528-7
|
| [36] |
J. L. Zhang, Z. W. Fang, H. W. Sun, Fast second-order evaluation for variable-order Caputo fractional derivative with applications to fractional sub-diffusion equations, Numer. Math. Theory Methods Appl., 15 (2022), 200–226. https://doi.org/10.4208/nmtma.oa-2021-0148 doi: 10.4208/nmtma.oa-2021-0148
|
| [37] |
X. C. Zheng, J Cheng, H. Wang, Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations, Inverse Probl., 35 (2019), 125002. https://doi.org/10.1088/1361-6420/ab3aa3 doi: 10.1088/1361-6420/ab3aa3
|
| [38] | V. Thom$\acute{\text{e}}$e, Galerkin Finite Element Methods for Parabolic Problems, Springer, 1984. https://doi.org/10.1007/bfb0071790 |
| [39] |
J. Y. Shen, F. H. Zeng, M. Stynes, Second-order error analysis of the averaged $L1$ scheme $\overline{L1}$ for time-fractional initial-value and subdiffusion problems, Sci. China Math., 67 (2024), 1641–1664. https://doi.org/10.1007/s11425-022-2078-4 doi: 10.1007/s11425-022-2078-4
|
| [40] |
X. Ye, J. Liu, B. Y. Zhang, H. F. Fu, Y. Liu, High order numerical methods based on quadratic spline collocation method and averaged L1 scheme for the variable-order time fractional mobile/immobile diffusion equation, Comput. Math. Appl., 171 (2024), 82–99. https://doi.org/10.1016/j.camwa.2024.07.009 doi: 10.1016/j.camwa.2024.07.009
|
| [41] | A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994. |
| [42] |
A. K. A. Khalifa, J. C. Eilbeck, Collocation with quadratic and cubic splines, IMA J. Numer. Anal., 2 (1982), 111–121. https://doi.org/10.1093/imanum/2.1.111 doi: 10.1093/imanum/2.1.111
|
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
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