scholarly journals Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Arshad Hussain ◽  
Marjan Uddin ◽  
Sirajul Haq ◽  
Hameed Ullah Jan
Keyword(s):  
Heat Equation ◽  
Numerical Solution ◽  
Meshless Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Cylindrical Coordinates ◽  
Runge Kutta Method ◽  
Radial Basis ◽  
Rectangular Coordinates ◽  
Meshless Method Of Lines

We propose a numerical solution to the heat equation in polar cylindrical coordinates by using the meshless method of lines approach. The space variables are discretized by multiquadric radial basis function, and time integration is performed by using the Runge-Kutta method of order 4. In radial basis functions (RBFs), much of the research are devoted to the partial differential equations in rectangular coordinates. This work is an attempt to explore the versatility of RBFs in nonrectangular coordinates as well. The results show that application of RBFs is equally good in polar cylindrical coordinates. Comparison with other cited works confirms that the present approach is accurate as well as easy to implement to problems in higher dimensions.

scholarly journals Meshless Method of Lines for Numerical Solution of Kawahara Type Equations

2011 ◽  
Vol 02 (05) ◽  
pp. 608-618 ◽  
Author(s):  
Nagina Bibi ◽  
Syed Ikram Abbas Tirmizi ◽  
Sirajul Haq
Keyword(s):  
Numerical Solution ◽  
Meshless Method ◽  
Method Of Lines ◽  
Meshless Method Of Lines

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ram Jiwari ◽  
Alf Gerisch
Keyword(s):  
Meshless Method ◽  
Order Of Convergence ◽  
Method Of Lines ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Second Order ◽  
Differential Quadrature ◽  
Time Dependent ◽  
Content Type ◽  
Radial Basis

Purpose This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time. Design/methodology/approach In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight. Findings The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems. Originality/value The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

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