Kansa RBF collocation method with auxiliary boundary centres for high order BVPs. (English) Zbl 1478.65135

Summary: In this study we apply a Kansa-radial basis function (RBF) collocation method to 2D and 3D boundary value problems (BVPs) governed by high order partial differential equations (PDEs) of order \(2\mathcal{N}\) where \(\mathcal{N}\in\mathbb{N}\), \(\mathcal{N}\geq 3\). As in such problems there are \(\mathcal{N}\) boundary conditions (BCs), \(\mathcal{N}\) distinct sets of boundary centres are needed. These could all be placed on the boundary with each set being different to the other or, alternatively, each set of boundary centres could be placed on a corresponding distinct curve surrounding the boundary of the problem. We apply these approaches to several 2D and 3D high order BVPs.


65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N85 Fictitious domain methods for boundary value problems involving PDEs
35J30 Higher-order elliptic equations
65D12 Numerical radial basis function approximation


Full Text: DOI


[1] Karageorghis, A.; Tappoura, D.; Chen, C. S., Kansa RBF method with auxiliary boundary centres for fourth order boundary value problems, Math. Comput. Simulation, 181, 581-597 (2021) · Zbl 07318236
[2] Tappoura, D., Kansa RBF Methods for the Solution of Second and Fourth Order Boundary Value Problems (2020), Department of Mathematics and Statistics, University of Cyprus, (M.Sc. thesis)
[3] Abdrabou, A.; El-Gamel, M., On the sinc-Galerkin method for triharmonic boundary-value problems, Comput. Math. Appl., 76, 520-533 (2018) · Zbl 1419.65130
[4] Bartezzaghi, A.; Dedè, L.; Quarteroni, A., Isogeometric analysis of high order partial differential equations on surfaces, Comput. Methods. Appl. Mech. Eng., 295, 446-469 (2015) · Zbl 1425.65145
[5] Gomez, H.; Nogueira, X., An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Eng., 249-253, 52-61 (2012) · Zbl 1348.74280
[6] Kapl, M.; Vitrih, V., A space of \(C^2 -\) smooth geometrically continuous isogeometric functions on two-patch geometries, Comput. Math. Appl., 73, 37-59 (2017) · Zbl 1368.65023
[7] Karageorghis, A., Efficient MFS algorithms for inhomogeneous polyharmonic problems, J. Sci. Comput., 46, 519-541 (2011) · Zbl 1270.65075
[8] Kirmani, S. K.N.; Jamil, R. N., Optimization of complex geometry using tenth order partial differential equation, Sci. Inquiry Rev., 2, 2, 23-31 (2018)
[9] Niiranen, J.; Kiendl, J.; Niemi, A. H.; Reali, A., Isogeometric analysis for sixth order boundary value problems of gradient-elastic kirchhoff plates, Comput. Methods Appl. Mech. Eng., 316, 328-348 (2017) · Zbl 1439.74037
[10] Satsanit, W., Solution to the triharmonic heat equation, Electron. J. Differential Equations, 2011, 4, 9 (2011) · Zbl 1223.46042
[11] Tagliabue, A.; Dedè, L.; Quarteroni, A., Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics, Comput. Fluids, 102, 277-303 (2014) · Zbl 1391.76360
[12] You, L. H.; Chang, J.; Yang, X. S.; Zhang, J. J., Solid modelling based on sixth order partial differential equations, Comput. Aided Des., 43, 720-729 (2011)
[13] Kansa, E. J., 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, 147-161 (1990) · Zbl 0850.76048
[14] Hon, Y. C.; Schaback, R., On unsymmetric collocation by radial basis functions, Appl. Math. Comput., 119, 2-3, 177-186 (2001) · Zbl 1026.65107
[15] Ling, L.; Opfer, R.; Schaback, R., Results on meshless collocation techniques, Eng. Anal. Bound. Elem., 30, 247-253 (2006) · Zbl 1195.65177
[16] Cavoretto, R.; De Rossi, A., An adapative LOOCV-based refinement scheme for RBF collocation methods over irregular domains, Appl. Math. Lett., 103, Article 106176 pp. (2020) · Zbl 1465.65158
[17] Agmon, S., Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc., 66, 77-80 (1960) · Zbl 0091.27301
[18] Barton, A.; Mayboroda, S., Boundary value problems for higher-order elliptic equations in non-smooth domains, (Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, Oper. Theory Adv. Appl, vol. 236 (2014), Birkhäuser/Springer: Birkhäuser/Springer Basel), 53-93 · Zbl 1326.35120
[19] John, F., (Partial Differential Equations. Partial Differential Equations, Applied Mathematical Sciences, vol. 1 (1978), Springer-Verlag: Springer-Verlag New York-Berlin) · Zbl 0426.35002
[20] http://blogs.mathworks.com/steve/2012/07/06/walking-along-a-path/.
[21] The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab.
[22] Li, M.; Chen, C. S.; Karageorghis, A., The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions, Comput. Math. Appl., 66, 2400-2424 (2013) · Zbl 1350.65136
[23] Fasshauer, G. E., (Meshfree Approximation Methods with MATLAB. Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, vol. 6 (2007), World Scientific Publishing Co. Pte. Ltd: World Scientific Publishing Co. Pte. Ltd Hackensack, NJ) · Zbl 1123.65001
[24] Jankowska, M. A.; Karageorghis, A.; Chen, C. S., Improved kansa RBF method for the solution of nonlinear boundary value problems, Eng. Anal. Bound. Elem., 87, 173-183 (2018) · Zbl 1403.65159
[25] Hardin, D. P.; Michaels, T.; Saff, E. B., A comparison of popular point configurations on \(\mathbb{S}^2\), Dolomites Res. Notes Approx., 9, 16-49 (2016) · Zbl 1370.31001
[26] Franke, R., Scattered data interpolation: tests of some methods, Math. Comp., 38, 181-200 (1982) · Zbl 0476.65005
[27] Kuo, L. H., On the Selection of a Good Shape Parameter for RBF Approximation and Its Applications for Solving PDEs (2015), University of Southern Mississippi, (Ph.D. Dissertation)
[28] Sarra, S.; Surgill, D., A random variable shape parameter strategy for radial basis function approaximation methods, Eng. Anal. Bound. Elem., 33, 1239-1245 (2009) · Zbl 1244.65192
[29] Xinag, S.; Wang, K. M.; Ai, Y. T.; Sha, Y. D.; Shi, H., Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation, Appl. Math. Model., 36, 1931-1938 (2012) · Zbl 1243.65023
[30] Jankowska, M. A.; Karageorghis, A., Variable shape parameter kansa RBF method for the solution of nonlinear boundary value problems, Eng. Anal. Bound. Elem., 103, 32-40 (2019) · Zbl 1464.65206
[31] Chen, W.; Hong, Y.; Lin, J., The sample solution approach for determination of the optimal shape parameter in multiquadric function of the kansa method, Comput. Math. Appl., 75, 2942-2954 (2018) · Zbl 1415.65264
[32] Biazar, J.; Hosami, M., An interval for the shape parameter in radial basis function approximation, Appl. Math. Comput., 315, 131-149 (2017) · Zbl 1426.65022
[33] Cavoretto, R.; De Rossi, A.; Mukhametzhanov, M. S.; Ya. D. Sergeyev, Y. D., On the search of the shape parameter in radial basis functions using univariate global optimixation methods, J. Global Optim., 79, 305-327 (2021) · Zbl 1470.65012
[34] Rippa, S., An algorithm for selecting a good value for the parameter \(c\) in radial basis function interpolation, Adv. Comput. Math., 11, 193-210 (1999) · Zbl 0943.65017
[35] Fasshauer, G. E.; Zhang, J. G., On choosing optimal shape parameters for RBF approximation, Numer. Algorithms, 45, 345-368 (2007) · Zbl 1127.65009
[36] Fu, Z.-J.; Reutskiy, S.; Sun, H.-G.; Ma, J.; Khan, M. A., A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains, Appl. Math. Lett., 94, 105-111 (2019) · Zbl 1411.65137
[37] Fu, Z.-J.; Zhang, J.; Li, P.-W.; Zheng, J.-H., A semi-Lagrangian meshless framework for numerical solutions of two-dimensional sloshing phenomenon, Eng. Anal. Bound. Elem., 12, 58-67 (2020) · Zbl 1464.65201
[38] Yang, F.; Yan, L.; Ling, L., Doubly stochastic radial basis function methods, J. Comput. Phys., 363, 87-97 (2018) · Zbl 1392.65016
[39] http://www.math.usm.edu/cschen/JCAM2021/Example1.m,Example6.m.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.