Radial basis function collocation method for an elliptic problem with nonlocal multipoint boundary condition. (English) Zbl 1403.65174
Summary: Radial basis function domain-type collocation method is applied for an elliptic partial differential equation with nonlocal multipoint boundary condition. A geometrically flexible meshless framework is suitable for imposing nonclassical boundary conditions which relate the values of unknown function on the boundary to its values at a discrete set of interior points. Some properties of the method are investigated by a numerical study of a test problem with the manufactured solution. Attention is mainly focused on the influence of nonlocal boundary condition. The standard collocation and least squares approaches are compared. In addition to its geometrical flexibility, the examined method seems to be less restrictive with respect to parameters of nonlocal conditions than, for example, methods based on finite differences.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Keywords:
elliptic problem; nonlocal multipoint boundary condition; meshless method; radial basis function; collocation; least squaresSoftware:
SciPyReferences:
| [1] | Day, W. A., Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Q Appl Math, 40, 319-330, (1985) · Zbl 0502.73007 |
| [2] | Day, W. A., Parabolic equations and thermodynamics, Q Appl Math, 50, 523-533, (1992) · Zbl 0794.35069 |
| [3] | Čiupaila, R.; Sapagovas, M.; Štikonienė, O., Numerical solution of nonlinear elliptic equation with nonlocal condition, Nonlinear Anal Model Control, 18, 4, 412-426, (2013) · Zbl 1291.65315 |
| [4] | Hazanee, A.; Lesnic, D., Determination of a time-dependent coefficient in the bioheat equation, Int J Mech Sci, 88, 259-266, (2014) |
| [5] | Díaz, J. I.; Padial, J. F.; Rakotoson, J. M., Mathematical treatment of the magnetic confinement in a current carrying stellarator, Nonlinear Anal, 34, 6, 857-887, (1998) · Zbl 0946.35119 |
| [6] | Huerta A, Belytschko T, Fernández-Méndez S, Rabczuk T. Meshfree methods. In: Stein E, de Borst R, Hughes TJR, editors. Encyclopedia of computational mechanics; vol. 1: Fundamentals, 2004. p. 279-309. doi:10.1002/0470091355.ecm005 |
| [7] | Liu GR. Meshfree methods: moving beyond the finite element method. 2nd ed. Boca Raton, FL: CRC Press; 2009. 10.1201/9781420082104. · Zbl 1205.74003 |
| [8] | Li H, Mulay SS. Meshless method and their numerical properties. Boca Raton, FL: CRC Press; 2013. 10.1201/b14492. · Zbl 1266.65001 |
| [9] | Buhmann MD. Radial basis functions: theory and implementations. Cambridge monographs on applied and computational mathematics, vol. 12. Cambridge: Cambridge University Press; 2003. doi:10.1017/CBO9780511543241 · Zbl 1038.41001 |
| [10] | Wendland H. Scattered data approximation. Cambridge monographs on applied and computational mathematics, vol. 17. Cambridge: Cambridge University Press; 2005. doi:10.1017/CBO9780511617539 · Zbl 1075.65021 |
| [11] | Sarra SA, Kansa EJ. Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations. Advances in computational mechanics, Duluth, GA: Tech Science Press; vol. 2. 2009. |
| [12] | Chen W, Fu ZJ, Chen CS. Recent advances in radial basis function collocation methods. Springer briefs in applied sciences and technology. Berlin, Heidelberg: Springer; 2014. doi:10.1007/978-3-642-39572-7 · Zbl 1282.65160 |
| [13] | Fornberg, B.; Flyer, N., Solving PDEs with radial basis functions, Acta Numer, 24, 215-258, (2015) · Zbl 1316.65073 |
| [14] | Fornberg B, Flyer N. A primer on radial basis functions with applications to the geosciences. CBMS-NSF regional conference series in applied mathematics; SIAM; 2015. doi:10.1137/1.9781611974041 · Zbl 1358.86001 |
| [15] | Dehghan, M.; Tatari, M., Use of radial basis functions for solving the second-order parabolic equation with nonlocal boundary conditions, Numer Methods Partial Differ Equ, 24, 3, 924-938, (2008) · Zbl 1143.65080 |
| [16] | Dehghan, M.; Tatari, M., On the solution of the non-local parabolic partial differential equations via radial basis functions, Appl Math Model, 33, 3, 1729-1738, (2009) · Zbl 1168.65403 |
| [17] | Dehghan, M.; Shokri, A., A meshless method for numerical solution of the one-dimensional wave equation with an integral conditions using radial basis functions, Numer Algorithm, 52, 3, 461-477, (2009) · Zbl 1178.65120 |
| [18] | Kazem, S.; Rad, J. A., Radial basis functions method for solving of a non-local boundary value problem with neumann׳s boundary conditions, Appl Math Model, 36, 6, 2360-2369, (2012) · Zbl 1246.65229 |
| [19] | Kadalbajoo, M. K.; Kumar, A.; Tripathi, L. P., A radial basis functions based finite differences method for wave equation with an integral condition, Appl Math Comput, 235, 8-16, (2015) · Zbl 1339.65122 |
| [20] | Yan, L.; Yang, F., The method of approximate particular solutions for the time-fractional diffusion equation with non-local boundary condition, Comput Math Appl, 70, 3, 254-264, (2015) · Zbl 1443.65256 |
| [21] | Ivanauskas, F.; Meškauskas, T.; Sapagovas, M., Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions, Appl Math Comput, 215, 7, 2716-2732, (2009) · Zbl 1191.65117 |
| [22] | Sapagovas, M.; Meškauskas, T.; Ivanauskas, F., Numerical spectral analysis of a difference operator with non-local boundary conditions, Appl Math Comput, 218, 14, 7515-7527, (2012) · Zbl 1246.65125 |
| [23] | Sajavičius, S., Stability of the weighted splitting finite-difference scheme for a two-dimensional parabolic equation with two nonlocal integral conditions, Comput Math Appl, 64, 11, 3485-3499, (2012) · Zbl 1268.65117 |
| [24] | Sajavičius, S., Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions—a case of two-dimensional Poisson equation, Eng Anal Bound Elem, 37, 4, 788-804, (2013) · Zbl 1297.65170 |
| [25] | Siraj-ul-Islam; Aziz, I.; Ahmad, M., Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions, Comput Math Appl, 69, 3, 180-205, (2015) · Zbl 1360.65290 |
| [26] | Sajavičius, S., Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions, Comput Math Appl, 67, 7, 1407-1420, (2014) · Zbl 1350.65132 |
| [27] | Sapagovas, M.; Čiupaila, R.; Jokšienė, Ž; Meškauskas, T., Computational experiment for stability analysis of difference schemes with nonlocal conditions, Informatica, 24, 2, 275-290, (2013) · Zbl 1360.65216 |
| [28] | Ashyralyev, A.; Ozturk, E., The numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition, Abstr Appl Anal, 2012, (2012), 13 p. Article ID 730804 · Zbl 1246.65196 |
| [29] | Ashyralyev, A.; Ozesenli Tetikoglu, FS, FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions, Abstr Appl Anal, 2012, (2012), 22 p. Article ID 454831 · Zbl 1261.65107 |
| [30] | Ashyralyev, A.; Ozesenli Tetikoglu, F. S., FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions, Math Methods Appl Sci, 37, 17, 2663-2676, (2014) · Zbl 1320.65150 |
| [31] | Volkov EA, Dosiyev AA. On the numerical solution of a multilevel nonlocal problem. Mediterr J Math 2016; in press. doi:10.1007/s00009-016-0704-x · Zbl 1359.65235 |
| [32] | Ashyralyev, A.; Gercek, O., Finite difference method for multipoint nonlocal elliptic-parabolic problems, Comput Math Appl, 60, 7, 2043-2052, (2010) · Zbl 1205.65230 |
| [33] | Ashyralyev, A.; Yildirim, O., Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator, Appl Math Comput, 218, 3, 1124-1131, (2011) · Zbl 1229.65147 |
| [34] | Yildirim, O.; Uzun, M., On the numerical solutions of high order stable difference schemes for the hyperbolic multipoint nonlocal boundary value problems, Appl Math Comput, 254, 210-218, (2015) · Zbl 1410.65333 |
| [35] | Ashyralyev, A.; Ozdemir, Y., On numerical solutions for hyperbolic-parabolic equations with the multipoint nonlocal boundary condition, J Frankl Inst, 351, 2, 602-630, (2014) · Zbl 1293.65112 |
| [36] | Jendele, L.; Červenka, J., On the solution of multi-point constraints—application to FE analysis of reinforced concrete structures, Comput Struct, 87, 15-16, 970-980, (2009) |
| [37] | Shirzadi, A., Solving 2D reaction-diffusion equations with nonlocal boundary conditions by the RBF-MLPG method, Comput Math Model, 25, 4, 521-529, (2014) · Zbl 1332.35175 |
| [38] | Cheng, A. H.D., Multiquadric and its shape parameter—a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation, Eng Anal Bound Elem, 36, 2, 220-239, (2012) · Zbl 1245.65162 |
| [39] | Berikelashvili, G.; Khomeriki, N., On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints, Nonlinear Anal Model Control, 19, 3, 367-381, (2014) · Zbl 1317.65216 |
| [40] | Jones E, Oliphant T, Peterson P, et al. SciPy: Open source scientific tools for Python. 2001. URL: 〈http://www.scipy.org/〉 |
| [41] | Sarra, S. A., Reguliarized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation, Eng Anal Bound Elem, 44, 76-86, (2014) · Zbl 1297.65032 |
| [42] | Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv Comput Math, 3, 3, 251-264, (1995) · Zbl 0861.65007 |
| [43] | Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv Comput Math, 11, 2-3, 193-210, (1999) · Zbl 0943.65017 |
| [44] | Scheuerer, M., An alternative procedure for selecting a good value for the parameter c in RBF-interpolation, Adv Comput Math, 34, 1, 105-126, (2011) · Zbl 1208.65022 |
| [45] | Uddin, M., On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method, Appl Math Model, 38, 1, 135-144, (2014) · Zbl 1427.65394 |
| [46] | Esmaeilbeigi, M.; Hosseini, M. M., A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by kansa׳s method, Appl Math Comput, 249, 419-428, (2014) · Zbl 1338.65254 |
| [47] | Kazem, S.; Hadinejad, F., PROMETHEE technique to select the best radial basis functions for solving the 2-dimensional heat equations based on Hermite interpolation, Eng Anal Bound Elem, 50, 29-38, (2015) · Zbl 1403.65093 |
| [48] | Golbabai, A.; Mohebianfar, E.; Rabiei, H., On the new variable shape parameter strategies for radial basis functions, Comput Appl Math, 34, 2, 691-704, (2015) · Zbl 1322.65040 |
| [49] | Štikonienė, O.; Sapagovas, M.; Čiupaila, R., On iterative methods for some elliptic equations with nonlocal conditions, Nonlinear Anal Model Control, 19, 3, 517-535, (2014) · Zbl 1316.65093 |
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