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

Arshad Hussain,Sirajul Haq,Marjan Uddin,Hameed Ullah Jan,Fazlollah Soleymani

doi:10.1155/2021/8862139

Abstract

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.

Highlights

We propose a meshless method of lines (MMOL) approach to study the heat equation in polar cylindrical coordinates
In the MQ method, the parameter c is called the shape parameter which shapes the stability and accuracy of the method. e selection of c requires much attention as there is a tradeoff between the stability and accuracy of the approximation method [19]. e MMOL has advantage over the traditional mesh-based methods [31]
We use radial basis functions (RBFs) to find numerical solution of the heat equation in the polar cylindrical form. e study explores the richness of RBF in polar cylindrical coordinates

Summary

Spectral radius

We consider numerical examples of the heat equation in polar cylindrical coordinates. Numerical solution was obtained at different numbers of nodes and different values of the shape parameter to show the convergence of the method. We utilized the exact solution (26) to derive the BCs. Various number of nodes and shape parameters were used in the computations. We consider equation (3) when a 1 over the region 0 ≤ ρ ≤ 1, − 1 ≤ z ≤ 1

Maximum error

Exact Numerical