Barycentric rational interpolation method for solving fractional cable equation

Jin Li; Yongling Cheng; Jin Li; Yongling Cheng

doi:10.3934/era.2023185

Electronic Research Archive

2023, Volume 31, Issue 6: 3649-3665. doi: 10.3934/era.2023185

Previous Article Next Article

Research article Special Issues

Barycentric rational interpolation method for solving fractional cable equation

Jin Li ,
Yongling Cheng ^,

School of Science, Shandong Jianzhu University, Jinan 250101, China

Received: 05 February 2023 Revised: 29 March 2023 Accepted: 06 April 2023 Published: 25 April 2023

A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.
- barycentric rational interpolation,
- collocation method,
- Fractional Cable equation,
- fractional derivative
Citation: Jin Li, Yongling Cheng. Barycentric rational interpolation method for solving fractional cable equation[J]. Electronic Research Archive, 2023, 31(6): 3649-3665. doi: 10.3934/era.2023185

Related Papers:

Abstract

A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.

References

[1]	Y. Lin, X. Li, C. Xu, Finite difference/spectral approximations for the fractional cable equation, Math. Comput., 80 (2011), 1369–1396. https://doi.org/10.1090/s0025-5718-2010-02438-x doi: 10.1090/s0025-5718-2010-02438-x
[2]	X. Hu, L. Zhang, Implicit compact difference schemes for the fractional cable equation, Appl. Math. Modell., 36 (2012), 4027–4043. https://doi.org/10.1016/j.apm.2011.11.027 doi: 10.1016/j.apm.2011.11.027
[3]	H. Zhang, X. Yang, X. Han, Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation, Comput. Math. Appl., 68 (2014), 1710–1722. https://doi.org/10.1016/j.camwa.2014.10.019 doi: 10.1016/j.camwa.2014.10.019
[4]	J. Quintana-Murillo, S. B. Yuste, An explicit numerical method for the Fractional Cable equation, Int. J. Differ. Equations, 2011 (2011), 1–12. https://doi.org/10.1155/2011/231920 doi: 10.1155/2011/231920
[5]	F. Liu, Q. Yang, I. Turner, Stability and convergence of two new implicit numerical methods for the Fractional Cable equation, Am. Soc. Mech. Eng., 2009 (2009), 1015–1024. https://doi.org/10.1115/DETC2009-86578 doi: 10.1115/DETC2009-86578
[6]	Y. Liu, Y. Du, H. Li, J. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dyn., 85 (2016), 2535–2548. https://doi.org/10.1007/s11071-016-2843-9 doi: 10.1007/s11071-016-2843-9
[7]	J. Liu, H. Li, Y. Liu, A new fully discrete finite difference/element approximation for fractional cable equation, J. Appl. Math. Comput., 52 (2016), 345–361. https://doi.org/10.1007/s12190-015-0944-0 doi: 10.1007/s12190-015-0944-0
[8]	P. Zhuang, F. Liu, V. Anh, I. Turner, The Galerkin finite element approximation of the fractional cable equation, in Proceedings of the 5th IFAC Symposium on Fractional Differentiation and Its Applications, 2012 (2012), 1–8.
[9]	X. Li, Theoretical analysis of the reproducing kernel gradient smoothing integration technique in Galerkin meshless Methods, J. Comput. Math., 41 (2023), 502–525. https://doi.org/10.4208/jcm.2201-m2021-0361 doi: 10.4208/jcm.2201-m2021-0361
[10]	J. Wan, X. Li, Analysis of a superconvergent recursive moving least squares approximation, Appl. Math. Lett., 133 (2022), 108223. https://doi.org/10.1016/j.aml.2022.108223 doi: 10.1016/j.aml.2022.108223
[11]	J. P. Berrut, S. A. Hosseini, G. Klein, The linear barycentric rational quadrature method for Volterra integral equations, SIAM J. Sci. Comput., 36 (2014), 105–123. https://doi.org/10.1137/120904020 doi: 10.1137/120904020
[12]	P. Berrut, G. Klein, Recent advances in linear barycentric rational interpolation, J. Comput. Appl. Math., 259 (2014), 95–107. https://doi.org/10.1016/j.cam.2013.03.044 doi: 10.1016/j.cam.2013.03.044
[13]	E. Cirillo, K. Hormann, On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes, J. Comput. Appl. Math., 349 (2019), 292–301. https://doi.org/10.1016/j.cam.2018.06.011 doi: 10.1016/j.cam.2018.06.011
[14]	M. Floater, H. Kai, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2007), 315–331. https://doi.org/10.1007/s00211-007-0093-y doi: 10.1007/s00211-007-0093-y
[15]	G. Klein, J. Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, SIAM J. Numer. Anal., 50 (2012), 643–656. https://doi.org/10.1137/110827156 doi: 10.1137/110827156
[16]	G. Klein, J. Berrut, Linear barycentric rational quadrature, BIT Numer. Math., 52 (2012), 407–424. https://doi.org/10.1007/s10543-011-0357-x doi: 10.1007/s10543-011-0357-x
[17]	J. Li, Y. Cheng, Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation, Comput. Appl. Math., 39 (2020). https://doi.org/10.1007/s40314-020-1114-z
[18]	J. Li, Y. Cheng, Linear barycentric rational collocation method for solving heat conduction equation, Numer. Methods Partial Differ. Equations, 37 (2021), 533–545. https://doi.org/10.1002/num.22539 doi: 10.1002/num.22539
[19]	J. Li, Y. Cheng, Barycentric rational method for solving biharmonic equation by depression of order, Numer. Methods Partial Differ. Equations, 37 (2021), 1993–2007. https://doi.org/10.1002/num.22638 doi: 10.1002/num.22638
[20]	J. Li, Linear barycentric rational collocation method for solving biharmonic equation, Demonstr. Math., 55 (2022), 587–603. https://doi.org/10.1515/dema-2022-0151 doi: 10.1515/dema-2022-0151
[21]	J. Li, X. Su, K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simul., 205 (2023), 340–367. https://doi.org/10.1016/j.matcom.2022.10.005 doi: 10.1016/j.matcom.2022.10.005
[22]	J. Li, Y. Cheng, Z. Li, Z. Tian, Linear barycentric rational collocation method for solving generalized Poisson equations, Math. Biosci. Eng., 20 (2023), 4782–4797. https://doi.org/10.3934/mbe.2023221 doi: 10.3934/mbe.2023221
[23]	J. Li, Barycentric rational collocation method for fractional reaction-diffusion equation, AIMS Math., 8 (2023), 9009–9026. https://doi.org/10.3934/math.2023451 doi: 10.3934/math.2023451
[24]	J. Li, Y. Cheng, Barycentric rational interpolation method for solving KPP equation, Electron. Res. Arch., 31 (2023), 3014–3029. https://doi.org/10.3934/era.2023152 doi: 10.3934/era.2023152
[25]	S. Li, Z. Wang, High Precision Meshless barycentric Interpolation Collocation Method–Algorithmic Program and Engineering Application. Science Publishing, Beijing, 2012.
[26]	Z. Wang, S. Li, Barycentric Interpolation Collocation Method for Nonlinear Problems, National Defense Industry Press, Beijing, 2015.
[27]	Z. Wang, Z. Xu, J. Li, Mixed barycentric interpolation collocation method of displacement-pressure for incompressible plane elastic problems, Chin. J. Appl. Mech., 35 (2018), 195–201.
[28]	Z. Wang, L. Zhang, Z. Xu, J. Li, Barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems, Chin. J. Appl. Mech., 35 (2018), 304–309.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)