Research article Special Issues

Lagrange radial basis function collocation method for boundary value problems in $ 1 $D

  • Received: 08 June 2023 Revised: 17 August 2023 Accepted: 27 August 2023 Published: 27 September 2023
  • MSC : 65F05, 65M30, 65N22

  • This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.

    Citation: Kawther Al Arfaj, Jeremy Levesly. Lagrange radial basis function collocation method for boundary value problems in $ 1 $D[J]. AIMS Mathematics, 2023, 8(11): 27542-27572. doi: 10.3934/math.20231409

    Related Papers:

  • This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.



    加载中


    [1] B. Fornberg, N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215–258. https://doi.org/10.1017/S0962492914000130 doi: 10.1017/S0962492914000130
    [2] M. Nawaz Khan, I. Ahmad, H. Ahmad, A radial basis function collocation method for space-dependent inverse heat problems, J. Appl. Comput. Mech., 6 (2020), 1187–1199. https://doi.org/10.22055/JACM.2020.32999.2123 doi: 10.22055/JACM.2020.32999.2123
    [3] C. S. Chen, A. Karageorghis, F. Dou, A novel RBF collocation method using fictitious centres, Appl. Math. Lett., 101 (2020), 106069. https://doi.org/10.1016/j.aml.2019.106069 doi: 10.1016/j.aml.2019.106069
    [4] R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76 (1971), 1905–1915. https://doi.org/10.1029/JB076i008p01905 doi: 10.1029/JB076i008p01905
    [5] W. R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl., 24 (1992), 121–138. https://doi.org/10.1016/0898-1221(92)90175-H doi: 10.1016/0898-1221(92)90175-H
    [6] S. Hubbert, T. M. Morton, Lp-error estimates for radial basis function interpolation on the sphere, J. Approx. Theory, 129 (2004), 58–77. https://doi.org/10.1016/j.jat.2004.04.006 doi: 10.1016/j.jat.2004.04.006
    [7] E. J. Kansa, Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–I surface approximations and partial derivative estimates, Comput. Math. Appl., 19 (1990), 127–145. https://doi.org/10.1016/0898-1221(90)90270-T doi: 10.1016/0898-1221(90)90270-T
    [8] C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11–22. https://doi.org/10.1007/BF01893414 doi: 10.1007/BF01893414
    [9] E. J. Kansa, Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19 (1990), 147–161. https://doi.org/10.1016/0898-1221(90)90271-K doi: 10.1016/0898-1221(90)90271-K
    [10] H. Y. Hu, Z. C. Li, A. H. D. Cheng, Radial basis collocation methods for elliptic boundary value problems, Comput. Math. Appl., 50 (2005), 289–320. https://doi.org/10.1016/j.camwa.2004.02.014 doi: 10.1016/j.camwa.2004.02.014
    [11] W. Du Toit, Radial basis function interpolation, Stellenbosch: Stellenbosch University, 2008.
    [12] M. D. Buhmann, Radial basis functions, Acta Numer., 9 (2000), 1–38. https://doi.org/10.1017/S0962492900000015.
    [13] F. Dell'Accio, F. Di Tommaso, O. Nouisser, N. Siar, Solving Poisson equation with Dirichlet conditions through multinode Shepard operators, Comput. Math. Appl., 98 (2021), 254–260. https://doi.org/10.1016/j.camwa.2021.07.021 doi: 10.1016/j.camwa.2021.07.021
    [14] F. Dell'Accio, F. Di Tommaso, G. Ala, E. Francomano, Electric scalar potential estimations for non-invasive brain activity detection through multinode Shepard method, In: IEEE 21st mediterranean electrotechnical conference (MELECON), Italy, 2022, 1264–1268. https://doi.org/10.1109/MELECON53508.2022.9842881
    [15] X. Li, S. Li, Meshless Galerkin analysis of the generalized Stokes problem, Comput. Math. Appl., 144 (2023), 164–181. https://doi.org/10.1016/j.camwa.2023.05.027 doi: 10.1016/j.camwa.2023.05.027
    [16] G. E. Fasshauer, Meshfree approximation methods with matlab, World Scientific, 2007. https://doi.org/10.1142/6437
    [17] R. Schaback, A practical guide to radial basis functions, 2007.
    [18] A. C. Faul, Iterative techniques for radial basis function interpolation, University of Cambridge, 2000. https://doi.org/10.17863/CAM.74829
    [19] H. Wendland, Scattered data approximation, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511617539
    [20] N. J. Higham, Functions of matrices: theory and computation, Society for Industrial and Applied Mathematics, 2008.
    [21] W. Chen, Z. J. Fu, C. S. Chen, Recent advances in radial basis function collocation methods, Heidelberg: Springer, 2014. https://doi.org/10.1007/978-3-642-39572-7
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(840) PDF downloads(50) Cited by(0)

Article outline

Figures and Tables

Figures(2)  /  Tables(17)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog