Meshless Galerkin method based on RBFs and reproducing Kernel for quasi-linear parabolic equations with dirichlet boundary conditions
Abstract
The main aim of this paper is to present a hybrid scheme of both meshless Galerkin and reproducing kernel Hilbert space methods. The Galerkin meshless method is a powerful tool for solving a large class of multi-dimension problems. Reproducing kernel Hilbert space method is an extremely efficient approach to obtain an analytical solution for ordinary or partial differential equations appeared in vast areas of science and engineering. The error analysis and convergence show that the proposed mixed method is very efficient. Since the solution space spanned by radial basis functions do not directly satisfy essential boundary conditions, an auxiliary parameterized technique is employed. Theoretical studies indicate that this new method is very stable, though a parameterized problem is employed instead of the main problem.
Keyword : Galerkin meshless method, radial basis functions, reproducing kernel Hilbert space method, quasi-linear parabolic equations
How to Cite
Mesrizadeh, M., & Shanazari, K. (2021). Meshless Galerkin method based on RBFs and reproducing Kernel for quasi-linear parabolic equations with dirichlet boundary conditions. Mathematical Modelling and Analysis, 26(2), 318-336. https://doi.org/10.3846/mma.2021.11436
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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O.A. Arqub and M. Al-Smadi. AtanganaBaleanu fractional approach to the solutions of BagleyTorvik and Painlev´e equations in Hilbert space. Chaos, Solitons & Fractals, 117:161–167, 2018. ISSN 0960-0779. https://doi.org/10.1016/j.chaos.2018.10.013
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O.A. Arqub and B. Maayah. Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – fractional Volterra integro-differential equations. Chaos, Solitons & Fractals, 126:394–402, 2019. ISSN 0960-0779. https://doi.org/10.1016/j.chaos.2019.07.023
P. Assari and M. Dehghan. A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions. Applied Mathematics and Computation, 315:424–444, 2017. ISSN 0096-3003. https://doi.org/10.1016/j.amc.2017.07.073
K. Atkinson and W. Han. Theoretical numerical analysis. Berlin, Springer, 2005. https://doi.org/10.1007/978-0-387-28769-0
V. Barbu. Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer-Verlag, New York, 2005. https://doi.org/10.1007/978-1-4419-5542-5
M.D. Buhmann. Radial basis functions. Acta Numerica, 9:1–38, 2000. https://doi.org/10.1017/S0962492900000015
Z. Cai. Convergence and error estimates for meshless Galerkin methods. Applied Mathematics and Computation, 184(2):908–916, 2007. ISSN 0096-3003. https://doi.org/10.1016/j.amc.2006.05.194
Z. Cai. Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem. Applied Mathematics and Computation, 215(6):2149–2153, 2009. ISSN 0096-3003. https://doi.org/10.1016/j.amc.2009.08.027
W. Chen, Z.J. Fu and C.S. Chen. Recent advances in radial basis function collocation methods. Springer, Berlin, 2014. https://doi.org/10.1007/978-3-642-39572-7
A. Corrigan, J. Wallin and Th. Wanner. A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data. arXiv, 01 2008.
M. Cui and Y. Lin. Nonlinear Numerical Analysis in Reproducing Kernel Space. Nova Science Publishers, Inc., USA, 2009. ISBN 1604564687.
M. Dehghan, M. Abbaszadeh and A. Mohebbi. Analysis of a meshless method for the time fractional diffusion-wave equation. Numerical Algorithms, 73(2):445– 476, 2016. https://doi.org/10.1007/s11075-016-0103-1
M. Dehghan and V. Mohammadi. The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The CrankNicolson scheme and the method of lines (MOL). Computers & Mathematics with Applications, 70(10):2292–2315, 2015. ISSN 0898-1221. https://doi.org/10.1016/j.camwa.2015.08.032
M. Dehghan and A. Shokri. A numerical method for solution of the twodimensional sine-Gordon equation using the radial basis functions. Mathematics and Computers in Simulation, 79(3):700–715, 2008. ISSN 0378-4754. https://doi.org/10.1016/j.matcom.2008.04.018
Y. Duan. A note on the meshless method using radial basis functions. Computers & Mathematics with Applications, 55(1):66–75, 2008. ISSN 0898-1221. https://doi.org/10.1016/j.camwa.2007.03.011
Y. Duan and Y.-J. Tan. A meshless Galerkin method for Dirichlet problems using radial basis functions. Journal of Computational and Applied Mathematics, 196(2):394–401, 2006. ISSN 0377-0427. https://doi.org/10.1016/j.cam.2005.09.018
W. Rudin. Principles of mathematical analysis. McGraw-hill, New York, 1964.
W. Rudin. Functional Analysis. International Series in Pure & Applied Mathematics, 1991.
R. Salehi and M. Dehghan. A generalized moving least square reproducing kernel method. Journal of Computational and Applied Mathematics, 249:120–132, 2013. ISSN 0377-0427. https://doi.org/10.1016/j.cam.2013.02.005
H. Wendland. Meshless Galerkin methods using radial basis functions. Mathematics of Computation of the American Mathematical Society, 68(228):1521– 1531, 1999. https://doi.org/10.1090/S0025-5718-99-01102-3
H. Wendland. Frontmatter, pp. i–iv. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2004.