Meshless collocation methods for time-dependent nonlocal problems based on radial basis functions

Qiao Zhuang; Yanzhi Zhang; Zhongqiang Zhang; Qiao Zhuang; Yanzhi Zhang; Zhongqiang Zhang

doi:10.3934/era.2026052

Electronic Research Archive

2026, Volume 34, Issue 2: 1124-1156. doi: 10.3934/era.2026052

Previous Article Next Article

Research article Special Issues

Meshless collocation methods for time-dependent nonlocal problems based on radial basis functions

1.
School of Science and Engineering, University of Missouri-Kansas City, Kansas City, MO 64110, USA
2.
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
3.
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received: 27 November 2025 Revised: 17 January 2026 Accepted: 27 January 2026 Published: 05 February 2026

We present radial basis function (RBF) collocation methods for time-dependent space-fractional problems on general bounded domains. Building on a recently developed approach for accurately computing the integral fractional Laplacian of any RBF, we design collocation schemes for fractional heat and Stokes equations using extended-domain techniques. In particular, we propose a numerical Leray projection method for fractional Stokes problems, where both the discrete projection operator and the collocation scheme are formulated on extended domains to handle complex domains. Numerical results demonstrate the effectiveness of the proposed methods in solving time-dependent nonlocal problems on complex domains.
- meshless collocation methods,
- time-dependent nonlocal problems,
- fractional heat equation,
- fractional Stokes equation,
- radial basis functions
Citation: Qiao Zhuang, Yanzhi Zhang, Zhongqiang Zhang. Meshless collocation methods for time-dependent nonlocal problems based on radial basis functions[J]. Electronic Research Archive, 2026, 34(2): 1124-1156. doi: 10.3934/era.2026052

Related Papers:

Abstract

We present radial basis function (RBF) collocation methods for time-dependent space-fractional problems on general bounded domains. Building on a recently developed approach for accurately computing the integral fractional Laplacian of any RBF, we design collocation schemes for fractional heat and Stokes equations using extended-domain techniques. In particular, we propose a numerical Leray projection method for fractional Stokes problems, where both the discrete projection operator and the collocation scheme are formulated on extended domains to handle complex domains. Numerical results demonstrate the effectiveness of the proposed methods in solving time-dependent nonlocal problems on complex domains.

References

[1]	K. Burrage, N. Hale, D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2145–A2172. https://doi.org/10.1137/110847007 doi: 10.1137/110847007
[2]	L. Vázquez, J. J. Trujillo, M. Pilar Velasco, Fractional heat equation and the second law of thermodynamics, Fract. Calc. Appl. Anal., 14 (2011), 334–342. https://doi.org/10.2478/s13540-011-0021-9 doi: 10.2478/s13540-011-0021-9
[3]	Asifa, T. Anwar, P. Kumam, M. Y. Almusawa, S. A. Lone, P. Suttiarporn, Exact solutions via prabhakar fractional approach to investigate heat transfer and flow features of hybrid nanofluid subject to shape and slip effects, Sci. Rep., 13 (2023), 7810. https://doi.org/10.1038/s41598-023-34259-9 doi: 10.1038/s41598-023-34259-9
[4]	M. Gunzburger, N. Jiang, F. Xu, Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion, Comput. Math. Appl., 75 (2018), 1973–2001. https://doi.org/10.1016/j.camwa.2017.06.035 doi: 10.1016/j.camwa.2017.06.035
[5]	B. P. Epps, B. Cushman-Roisin, Turbulence modeling via the fractional Laplacian, preprint, arXiv: 1803.05286.
[6]	S. Hittmeir, S. Merino-Aceituno, Kinetic derivation of fractional Stokes and Stokes-Fourier systems, Kinet. Rel. Models, 9 (2016), 105–129. https://doi.org/10.3934/krm.2016.9.105 doi: 10.3934/krm.2016.9.105
[7]	S. Lin, M. Azaïez, C. Xu, A fractional Stokes equation and its spectral approximation, Int. J. Numer. Anal. Model., 15 (2018), 170–192.
[8]	H. Wang, T. S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444–A2458. https://doi.org/10.1137/12086491X doi: 10.1137/12086491X
[9]	K. Sayevand, J. T. Machado, V. Moradi, A new non-standard finite difference method for analyzing the fractional Navier-Stokes equations, Comput. Math. Appl., 78 (2019), 1681–1694. https://doi.org/10.1016/j.camwa.2018.12.016 doi: 10.1016/j.camwa.2018.12.016
[10]	X. Li, X. You, Mixed finite element methods for fractional Navier-Stokes equations, J. Comput. Math., 39 (2021), 130–146. https://doi.org/10.4208/jcm.1911-m2018-0153 doi: 10.4208/jcm.1911-m2018-0153
[11]	Y. Jiao, T. Li, Z. Zhang, Jacobi spectral collocation method of space-fractional Navier-Stokes equations, Appl. Math. Comput., 488 (2025), 129111. https://doi.org/10.1016/j.amc.2024.129111 doi: 10.1016/j.amc.2024.129111
[12]	A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numer. Math., 54 (2014), 937–954. https://doi.org/10.1007/s10543-014-0484-2 doi: 10.1007/s10543-014-0484-2
[13]	J. A. Rosenfeld, S. A. Rosenfeld, W. E. Dixon, A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions, J. Comput. Phys., 390 (2019), 306–322. https://doi.org/10.1016/j.jcp.2019.02.015 doi: 10.1016/j.jcp.2019.02.015
[14]	J. Burkardt, Y. Wu, Y. Zhang, A unified meshfree pseudospectral method for solving both classical and fractional PDEs, SIAM J. Sci. Comput., 43 (2021), A1389–A1411. https://doi.org/10.1137/20M1335959 doi: 10.1137/20M1335959
[15]	Y. Wu, Y. Zhang, Variable-order fractional laplacian and its accurate and efficient computations with meshfree methods, J. Sci. Comput., 99 (2024), 18. https://doi.org/10.1007/s10915-024-02472-x doi: 10.1007/s10915-024-02472-x
[16]	Z. Hao, Z. Cai, Z. Zhang, Fractional-order dependent radial basis functions meshless methods for the integral fractional laplacian, Comput. Math. Appl., 178 (2025), 197–213. https://doi.org/10.1016/j.camwa.2024.11.027 doi: 10.1016/j.camwa.2024.11.027
[17]	X. G. Zhu, Y. F. Nie, Z. H. Ge, Z. B. Yuan, J. G. Wang, A class of RBFs-based DQ methods for the space-fractional diffusion equations on 3d irregular domains, Comput. Mech., 66 (2020), 221–238. https://doi.org/10.1007/s00466-020-01848-8 doi: 10.1007/s00466-020-01848-8
[18]	J. Liu, X. Li, X. Hu, A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation, J. Comput. Phys., 384 (2019), 222–238. https://doi.org/10.1016/j.jcp.2018.12.043 doi: 10.1016/j.jcp.2018.12.043
[19]	B. Fornberg, C. Piret, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30 (2008), 60–80. https://doi.org/10.1137/060671991 doi: 10.1137/060671991
[20]	B. Fornberg, E. Larsson, N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33 (2011), 869–892. https://doi.org/10.1137/09076756X doi: 10.1137/09076756X
[21]	E. Hanert, C. Piret, Numerical solution of the space-time fractional diffusion equation: Alternatives to finite differences, in Proceedings of the 5th Symposium on Fractional Differentiation and its Applications, 2012.
[22]	R. Cavoretto, A. De Rossi, M. Donatelli, S. Serra-Capizzano, Spectral analysis and preconditioning techniques for radial basis function collocation matrices, Numer. Linear Algebra Appl., 19 (2012), 31–52. https://doi.org/10.1002/nla.774 doi: 10.1002/nla.774
[23]	S. Kumar, C. Piret, Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials, Appl. Numer. Math., 143 (2019), 300–315. https://doi.org/10.1016/j.apnum.2019.04.012 doi: 10.1016/j.apnum.2019.04.012
[24]	Q. Zhuang, A. Heryudono, F. Zeng, Z. Zhang, Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions, Appl. Math. Comput., 469 (2024), 128548. https://doi.org/10.1016/j.amc.2024.128548 doi: 10.1016/j.amc.2024.128548
[25]	M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7–51. https://doi.org/10.1515/fca-2017-0002 doi: 10.1515/fca-2017-0002
[26]	N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren der mathematischen Wissenschaftern, Springer, 1972.
[27]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
[28]	M. Mori, M. Sugihara, The double-exponential transformation in numerical analysis, J. Comput. Appl. Math., 127 (2001), 287–296. https://doi.org/10.1016/S0377-0427(00)00501-X doi: 10.1016/S0377-0427(00)00501-X
[29]	E. J. Fuselier, V. Shankar, G. B. Wright, A high-order radial basis function (RBF) Leray projection method for the solution of the incompressible unsteady Stokes equations, Comput. Fluids, 128 (2016), 41–52. https://doi.org/10.1016/j.compfluid.2016.01.009 doi: 10.1016/j.compfluid.2016.01.009
[30]	E. J. Fuselier, G. B. Wright, A radial basis function method for computing Helmholtz–Hodge decompositions, IMA J. Numer. Anal., 37 (2017), 774–797. https://doi.org/10.1093/imanum/drw027 doi: 10.1093/imanum/drw027
[31]	H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9780511617539.002
[32]	T. Gneiting, Strictly and non-strictly positive definite functions on spheres, Bernoulli, 19 (2013), 1327–1349. https://doi.org/10.3150/12-BEJSP06 doi: 10.3150/12-BEJSP06
[33]	C. Foias, O. Manley, R. Rosa, R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511546754
[34]	H. Helmholtz, Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen, J. Reine Angew. Math., 55 (1858), 25–55. https://doi.org/10.1515/crll.1858.55.25 doi: 10.1515/crll.1858.55.25
[35]	H. Wendland, Divergence-free kernel methods for approximating the Stokes problem, SIAM J. Numer. Anal., 47 (2009), 3158–3179. https://doi.org/10.1137/080730299 doi: 10.1137/080730299
[36]	A. J. Chorin, J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer New York, 1993. https://doi.org/10.1007/978-1-4612-0883-9
[37]	J. Neustupa, P. Penel, The Navier-Stokes equation with slip boundary conditions, RIMS Kokyuroku, 1536 (2007), 46–57.
[38]	R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, Society for Industrial and Applied Mathematics, 2007. https://doi.org/10.1137/1.9780898717839
[39]	I. Akramov, S. Götschel, M. Minion, D. Ruprecht, R. Speck, Spectral deferred correction methods for second-order problems, SIAM J. Sci. Comput., 46 (2024), A1690–A1713. https://doi.org/10.1137/23M1592596 doi: 10.1137/23M1592596
[40]	J. Wang, X. Chen, J. Chen, A high-precision numerical method based on spectral deferred correction for solving the time-fractional Allen-Cahn equation, Comput. Math. Appl., 180 (2025), 1–27. https://doi.org/10.1016/j.camwa.2024.11.034 doi: 10.1016/j.camwa.2024.11.034
[41]	R. Glowinski, Y. A. Kuznetsov, Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems, Comput. Methods Appl. Mech. Eng., 196 (2007), 1498–1506. https://doi.org/10.1016/j.cma.2006.05.013 doi: 10.1016/j.cma.2006.05.013
[42]	C. Garoni, C. Manni, F. Pelosi, H. Speleers, Study and use of spectral symbol properties for isogeometric matrices on trimmed geometries, Numer. Linear Algebra Appl., 32 (2025), e2601. https://doi.org/10.1002/nla.2601 doi: 10.1002/nla.2601
[43]	M. Mazza, S. Serra-Capizzano, R. L. Sormani, Algebra preconditionings for 2D Riesz distributed-order space-fractional diffusion equations on convex domains, Numer. Linear Algebra Appl., 31 (2024), e2536. https://doi.org/10.1002/nla.2536 doi: 10.1002/nla.2536
[44]	Y. Wu, Y. Zhang, A universal solution scheme for fractional and classical PDEs, preprint, arXiv: 2102.00113.

Reader Comments

Your name:*

Email:*
© 2026 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)