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Mathematics 2019, 7(3), 216; https://doi.org/10.3390/math7030216

An Efficient Local Formulation for Time–Dependent PDEs

1
Department of Mathematics, University of Swabi, Swabi 23430, Pakistan
2
Department of Basic Sciences, University of Engineering and Technology, Peshawar 25000, Pakistan
3
Department of Basic Sciences, CECOS University of IT and Emerging Sciences, Peshawar 25000, Pakistan
4
KMUTT-Fixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
5
KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
*
Author to whom correspondence should be addressed.
Received: 30 January 2019 / Revised: 18 February 2019 / Accepted: 20 February 2019 / Published: 26 February 2019
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Abstract

In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains. View Full-Text
Keywords: local meshless method; RBFs; irregular domains; Kortewege-de Vries types equations; reaction-diffusion Brusselator system local meshless method; RBFs; irregular domains; Kortewege-de Vries types equations; reaction-diffusion Brusselator system
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ahmad, I.; Ahsan, M.; Din, Z.-U.; Masood, A.; Kumam, P. An Efficient Local Formulation for Time–Dependent PDEs. Mathematics 2019, 7, 216.

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