Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation

Kedir Aliy,Hailu Muleta,Alemayehu Shiferaw

doi:10.11648/j.ajmcm.20210602.12

Abstract

In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E<sup>2</sup>), maximum absolute error (E<sup>∞</sup>) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.

Highlights

Partial Differential Equations (PDEs) are mathematical equations that are significant in modeling physical phenomena that occur in nature
The condition number of the system matrix of the present method is in the range 4.0762E+07 ≤ κ ( A) ≤ 7.3104E+17 whereas the condition number of the system matrix presented by Tatari and Dehghan, 20103.0267E+15 ≤ κ ( A) ≤ 1.9676E+19
The effect of the condition number on the accuracy of the numerical solution is more significant on the method presented by Tatari and Dehghan, than on the numerical solution of the present method

Summary

Introduction

Partial Differential Equations (PDEs) are mathematical equations that are significant in modeling physical phenomena that occur in nature. Applications of PDEs can be found in physics, engineering, mathematics, and finance. PDEs have a wide range of applications to real world problems in science and engineering, the majority of PDEs do not have analytical solutions. Mesh-dependent methods such as the finite difference method (FDM), finite element method (FEM), and boundary element method (BEM) have been used to solve PDEs [16]. Despite their great success in the past decades in many branches of science and engineering, these mesh-based methods require meshes or grids as the solution domain. May be the complications of these methods include a slow rate of convergence, spatial dependence, instability, low accuracy, and difficulty of implementation in complex geometries [16, 18]

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Conclusion