A Numerical Approach based on Differential Quadrature Method for Nonlinear Heat Equation

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Mathematics, Statistics & Computer Sciences, Semnan, Iran;

2 Department of Mathematics, Faculty of Mathematics, Statistics & Computer Sciences, Semnan, Iran

Abstract

In this research paper, a numerical method for one- and two- dimensional heat equation with nonlinear diffusion conductivity and source terms is proposed. In this work, the numerical technique is based on the polynomial differential quadrature method for discretization of the spatial domain. The resulting nonlinear system time depending ordinary differential equations is discretized by using the second order Runge–Kutta methods. The Chebyshev-Gauss-Lobatto points in this paper are used for collocation points in spatial discretization. We study accuracy in terms of L_∞ error norm and maximum absolute error along time levels. Finally, several test examples demonstrate the accuracy and efficiency of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear diffusion equations.

Keywords

References

1. S. Abbasbandy, E. Shivanian, Exact analytical solution of a nonlinear equation arising in heat transfer, Physics Letters A, 374(4), 567–574 (2010).
2. C. Arslanturk, A decomposition method for fin effciency of convective straight fins with temperature-dependent thermal conductivity, International communications in heat and mass transfer, 32(6), 831–841 (2005).
3. A. Moradi, H. Ahmadikia, Analytical solution for different profiles of fin with temperature-dependent thermal conductivity, Mathematical Problems in Engineering, 2010, (2010).
4. A. Aziz, F. Khani, Convection–radiation from a continuously moving fin of variable thermal conductivity, Journal of the Franklin Institute, 348(4), 640–651 (2011).
5. D. D. Ganji, Z. G. Ziabkhsh, D. H. Ganji, Determination of temperature distribution for annular fins with temperature dependent thermal conductivity by HPM, Thermal science, 15(1), 111–115 (2011). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 8, Issue 1
November 2023
Pages 65-79

Files

History

  • Receive Date: 21 May 2024
  • Revise Date: 11 June 2024
  • Accept Date: 14 June 2024

Share

How to cite

Statistics