Numerical simulation of fractional power diffusion biosensors
Abstract
The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.
Keyword : mathematical modelling, diffusion-reaction equations, fractional power of elliptic operators, finite volume schemes, splitting method, biosensors
How to Cite
Dapšys, I., & Čiegis, R. (2023). Numerical simulation of fractional power diffusion biosensors. Mathematical Modelling and Analysis, 28(2), 180–193. https://doi.org/10.3846/mma.2023.17583
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P.N. Vabishchevich. Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operator. Applied Numerical Mathematics, 165:414–430, 7 2021. https://doi.org/10.1016/j.apnum.2021.03.006
H. Zhang, X. Jiang, F. Zeng and G.E. Karniadakis. A stabilized semiimplicit Fourier spectral method for nonlinear space-fractional reactiondiffusion equations. Journal of Computational Physics, 405:109141, 3 2020. https://doi.org/10.1016/j.jcp.2019.109141
A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otárola and A.J. Salgado. Numerical methods for fractional diffusion. Computing and Visualization in Science, 19(5):19–46, 12 2018. https://doi.org/10.1007/s00791-018-0289-y
A. Bonito and J.E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Mathematics of Computation, 84(295):2083–2110, 3 2015. https://doi.org/10.1090/s0025-5718-2015-02937-8
A. Bueno-Orovio, D. Kay and K. Burrage. Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numerical Mathematics, 54(4):937–954, 12 2014. https://doi.org/10.1007/s10543-014-0484-2
R. Čiegis and Rem. Čiegis. Numerical algorithms for solving the optimal control problem of simple bioreactors. Nonlinear Analysis: Modelling and Control, 24(4):626–638, 6 2019. https://doi.org/10.15388/NA.2019.4.8
R. Čiegis, Rem. Čiegis and I. Dapšys. A comparison of discrete schemes for numerical solution of parabolic problems with fractional power elliptic operators. Mathematics, 9(12), 6 2021. https://doi.org/10.3390/math9121344
R. Čiegis and I. Dapšys. On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type. Mathematics, 10(13):2155, 6 2022. https://doi.org/10.3390/math10132155
R. Čiegis, I. Dapšys and Rem. Čiegis. A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators. Axioms, 11(3):98, 2 2022. https://doi.org/10.3390/axioms11030098
R. Čiegis, P. Katauskis and V. Skakauskas. The robust finite-volume schemesˇ for modeling nonclassical surface reactions. Nonlinear Analysis: Modelling and Control, 23(2):234–250, 4 2018. https://doi.org/10.15388/NA.2018.2.6
R. Čiegis, G. Panasenko, K. Pileckas and V. Šumskas. ADI scheme for partially dimension reduced heat conduction models. Computers & Mathematics with Applications, 80(5):1275–1286, 2020. https://doi.org/10.1016/j.camwa.2020.06.012
R. Čiegis, O. Suboč and Rem. Čiegis. Numerical simulation of nonlocal delayed feedback controller for simple bioreactors. Informatica, 29(2):233–249, 1 2018. https://doi.org/10.15388/Informatica.2018.165
S. Harizanov, N. Kosturski, I. Lirkov, S. Margenov and Y. Vutov. Reduced multiplicative (BURA-MR) and additive (BURA-AR) best uniform rational approximation methods and algorithms for fractional elliptic equations. Fractal and Fractional, 5(3):61, 6 2021. https://doi.org/10.3390/fractalfract5030061
S. Harizanov, R. Lazarov, S. Margenov, P. Marinov and J. Pasciak. Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. Journal of Computational Physics, 408:109285, 5 2020. https://doi.org/10.1016/j.jcp.2020.109285
S. Harizanov, R. Lazarov, S. Margenov, P. Marinov and Y. Vutov. Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Numerical Linear Algebra with Applications, 25:e2167, 10 2018. https://doi.org/10.1002/nla.2167
S. Harizanov, S. Margenov, P. Marinov and Y. Vutov. Volume constrained 2phase segmentation method utilizing a linear system solver based on the best uniform polynomial approximation of x−1/2. Journal of Computational and Applied Mathematics, 310:115–128, 1 2017. https://doi.org/10.1016/j.cam.2016.06.020
C. Hofreither. A unified view of some numerical methods for fractional diffusion. Computers & Mathematics with Applications, 80(2):332–350, 7 2020. https://doi.org/10.1016/j.camwa.2019.07.025
C. Hofreither. An algorithm for best rational approximation based on barycentric rational interpolation. Numerical Algorithms, 88(1):365–388, 9 2021. https://doi.org/10.1007/s11075-020-01042-0
W. Hundsdorfer and J. Verwer. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Berlin, Heidelberg, 2003. https://doi.org/10.1007/978-3-662-09017-6
F. Ivanauskas, V. Laurinavičius, M. Sapagovas and A. Nečiporenko. Reaction–diffusion equation with nonlocal boundary condition subject to PIDcontrolled bioreactor. Nonlinear Analysis: Modelling and Control, 22(2):261– 272, 3 2017. https://doi.org/10.15388/NA.2017.2.8
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. Elsevier, Amsterdam, 2006.
H.G. Lee. A second-order operator splitting Fourier spectral method for fractional-in-space reaction–diffusion equations. Journal of Computational and Applied Mathematics, 333:395–403, 5 2018. https://doi.org/10.1016/j.cam.2017.09.007
R. Metzler and J. Klafter. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General, 37(31):161–208, 2004. https://doi.org/10.1088/0305-4470/37/31/R01
Y. Nakatsukasa, O. Séte and L. N. Trefethen. The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing, 40(3):A1494–A1522, 1 2018. https://doi.org/10.1137/16M1106122
R.H. Nochetto, E. Otárola and A.J. Salgado. A PDE approach to space-time fractional parabolic problems. SIAM Journal on Numerical Analysis, 54(2):848– 873, 3 2016. https://doi.org/10.1137/14096308X
I. Podlubny. Fractional differential equations, mathematics in science and engineering. Academic Press, New York, 1999.
P.N. Vabishchevich. Splitting schemes for non-stationary problems with a rational approximation for fractional powers of the operator. Applied Numerical Mathematics, 165:414–430, 7 2021. https://doi.org/10.1016/j.apnum.2021.03.006
H. Zhang, X. Jiang, F. Zeng and G.E. Karniadakis. A stabilized semiimplicit Fourier spectral method for nonlinear space-fractional reactiondiffusion equations. Journal of Computational Physics, 405:109141, 3 2020. https://doi.org/10.1016/j.jcp.2019.109141