The use of radial basis functions by variable shape parameter for solving partial differential equations

Document Type : research paper


1 Department of Applied Mathematics, Faculty of Basic Science, Islamic Azad University, Science and Research Branch, Tehran, Iran

2 Department of Applied Mathematics, Faculty of Basic Science , Imam Khomeini International University, Qazvin, Iran


In this paper, some meshless methods based on the local Newton basis functions are used to solve some time dependent partial differential equations. For stability reasons, used variably scaled radial kernels for constructing Newton basis functions. In continuation, with considering presented basis functions as trial functions, approximated solution functions in the event of spatial variable with collocation method.  Then, with aid of method of lines obtained a system of ordinary differential equations according to solution function in the event of time. Methods applied for solving the nonlinear Burgers’ equation and couple Burgers’ equation. The numerical results show that the proposed method is efficient, accurate and stable.


Article Title [فارسی]

استفاده از توابع پایه شعاعی با پارامتر شکلی متغیر برای حل معادلات دیفرانسیل جزیی

Authors [فارسی]

  • حنانه نوجوان 1
  • سعید عباسبندی 2
  • توفیق الهویرنلو 1
1 گروه ریاضی، دانشکده علوم پایه، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران
2 گروه ریاضی کاربردی، دانشکده علوم پایه، دانشگاه بین المللی امام خمینی (ره)، قزوین، ایران
Abstract [فارسی]

در این مقاله از روش­های بدون شبکه مبتنی بر توابع پایه­ای نیوتن موضعی برای حل معادلات دیفرانسیل با مشتقات جزیی وابسته به زمان استفاده شده است. به منظور پایداری بیشتر از هسته­های شعاعی به طور متغیر مقیاس شده برای ساخت توابع پایه­ای نیوتن استفاده شده است. در ادامه با در نظر گرفتن توابع پایه­ای معرفی شده به عنوان توابع آزمون، تابع جواب در راستای متغیر مکان با استفاده از توابع آزمون به روش هم مکانی تقریب زده می­شود. سپس با استفاده از روش خطوط، به دستگاهی از معادلات با مشتقات معمولی بر حسب تابع جواب در راستای متغیر زمان دست یافتیم. روش­های معرفی شده را برای حل معادله غیرخطی برگرز به کار گرفته و با مشاهده نتایج عددی دقت و کارآیی روش مشخص خواهد شد.

Keywords [فارسی]

  • روش بدون شبکه موضعی
  • روش خطوط
  • توابع پایه‌ای نیوتن
  • هسته شعاعی به طور متغیر مقیاس شده
  • معادله غیرخطی برگرز
[1] Belytschko, T., Lu, Y.Y., Gu, L. (1994).” Element-free Galerkin methods”. Int. J. Numer. Methods Engrg. 37, 229-256.
[2] Liu, W.K., Jun, S., Zhang, Y.F. (1995). ”Reproducing kernel particle
methods”. Int. J. Numer. Methods Fluids 21, 1081-1106.
[3] Kansa, E.J. (1990).”Multiquadrics–a scattered data approximationscheme with application to computational fluid dynamics, part I”. Comput. Math Appl. 19, 127–145.
 [4] Mukherjee, Y.X., Mukherjee, S. (1997). ”The boundary node method for potential problems”. Int. J. Numer. Methods Eng.40,797-815.
[5] Zhu, T., Zhang, J.D., Atluri, S.N.(1998).”A local boundary integralequation (LBIE) method in computational mechanics and a meshless discretization approach”. Comput. Mech. 21, 223-235.
[6] Sladek, J., Sladek, V., Atluri, S.N. (2000). ”Local boundary integral equation (LBIE) method for solving problem of elasticity with nonhomogeneous material properties”. Comput. Mech. 24, 456-462.
[7] Dehghan, M., Mirzaei, D. (2008). ”Numerical solution to the unsteady two-dimensional Schrodinger equation using meshless local boundary integral equation method”. Int. J. Numer. Methods Eng.76,501-520.
[8] Mohebbi, A., Dehghan, M. (2008). ” High order compact solution of the one-space-dimensional linear hyperbolic equation”. Numer Methods Partial Differential Equations 24, 1222-1235.
[9] Atluri, S.N., Shen,S.,"The Meshless Local Petrov-Galerkin Method, (MLPG)".Tech Science Press, 2002.
[10] Dehghan, M., Mirzaei, D. (2009).” Meshless Local Petrov-Galerki method (MLPG) for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity”. Appl. Numer.Math. 59, 1043-1058.
 [11] Dehghan, M., Mirzaei, D. (2008). ”The meshless local Petrov Galerkin (MLPG) method for the generalized two-dimensional nonlinear Schrodinger equation”. Engineering Analysis with Boundary Elements. 32, 747-756.
[12] Mazarei M.M., A. Aminataei A. (2012). ”Numerical Solution of Poisson’s Equation Using a Combination of Logarithmic and Multiquadric Radial Basis Function Networks”. Journal of Applied Mathematics.
 [13] Liu, G.R. (2003). ”Mesh Free Methods: Moving beyond the Finite Element Method”. CRC press.
[14]Nguyen, V.P., Rabczuk, T., Bordas, S.,
 Duflot, M. (2008). ”Meshless methods: A review and computer implementation aspects," Math. Comput. Simul. 79, 763-813.
[15] Sarler, B. (2007). ”From global to local radial basis function collocation method for transport phenomena”. Berlin: Springer, 257-282.
 [16] Yun, D.F., Hon, Y.C. (2016). ”Improved localized radial basis function collocation method for multi-dimensional convectiondominated problems”. Eng. Anal. Bound. Elem. 67, 63–80.
[17] Sarra, S. (2012). ”A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains". Appl. Math. Comp. 218, 9853–9865.
[18] Sarler, B. (2007). ”From global to local radial basis function collocation method for transport phenomena”. Berlin: Springer, 257-282.
 [19] Lee, C., Liu, X., Fan, S. (2003). ”Local multiquadric approximation for solving boundary value problems”. Comput. Mech. 30, 396–409.
[20] Abbasbandy, S., Darvishi, M.T. (2005). ”A numerical solution of Burgers’ equation by time discretization of Adomian’s  decomposition method”. Appl. Math. Comput. 170,95–102.
[21] Asaithambi, A. (2010).”Numerical solution of the Burgers’ equation by automatic differentiation”.Appl. Math. Comput. 216, 2700-2708.
 [22] Dag, I., Irk, D., Saka, B. (2015).”A numerical solution of the Burgers’equation using cubic B- splines”.  Appl. Math. Comput. 163(1),199-211.
[23] Hashemian, A. Shodja, H.M. (2008).”A meshless approach for solution of Burgers’ equation”. J. Comput. Appl. Math. 220, 226–239.
[24] Jiwari, R. (2012). ”A haar wavelet quasi linearization approach for numerical simulation of Burgers’ equation”. Comput. Phys. Commun.
[25] Jiwari, R. (2015). ”A hybrid numerical scheme for the numerical solution of the Burgers’ equation”. Comput. Phys. Commun. 188,59-67.,
[26] Mittal, R., Jain, R. (2012). ”Numerical solutions of nonlinear burgers equation with modified cubic b-splines collocation method". Appl. Math. Comput. 218(15), 7839–7855..
[27] Xie, H., Zhou, J., Jiang, Z., Guo, X. (2016). ”Approximations for Burgers’ equations with C-N scheme and RBF collocation methods". J. Nonlinear Sci. Appl. 9, 3727–3734.
[28] Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R. (2015). ”Interpolation with variably scaled kernels”. IMA J. Numer. Anal. 35,199-219.
 [29] L.T. Luh, The shape parameter in the Gaussian function, Comput. Math. Appl. 63 (2012) 687–694..
 [30] Sarra, S., Sturgill, D. (2009). ”A random variable shape parameter
strategy for radial basis function approximation methods”. Eng. Anal. Bound. Elem. 33, 1239–1245
[31] Muller, S., Schaback, R. (2009). ” A Newton basis for kernel spaces”. J. Approx. Theory. 161, 645–655.
[32] Pazouki, M., Schaback, R. (2011). Bases for kernel-based spaces". J. Comput. Appl. Math. 236, 575–588
[33] Schaback, R. (2011). ”Matlab Programming for Kernel -Based methods”. Technical Report, http://num.math.unigoettingen  de/schaback/research/papers/MPfKBM.pdf.
[34] Caldwell, J. Smith, P. (1982). ”Solution of Burgers’ equation with a large Reynolds number”. Appl. Math. Model. 6, 381–385
[35] Hon, Y.C., Mao, X.Z. (1998 )."    An efficient numerical scheme for Burgers’ equation”. Appl.Math.Comp.95,37-50.