Bernstein Series Solution of the Heat Equation in 2-D

Abstract

A broad class of steady-state physical problems can be reduced to finding the harmonic functions that satisfy certain boundary conditions. The Dirichlet problem for the Laplace equation is one of the above mentioned problems. In this paper, a numerical matrix method is developed for numerically solving the Heat equation in 2-D. The method converts the heat equation in 2-D to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis is included to demonstrate the validity and applicability of the technique. Finally, the effectiveness of the method is illustrated in the heat equation for a cut ring region.

Keywords

• Heat equation

References

1. Ahmadi MR, Adibi H. The Chebyshev tau technique for the solution of Laplace’s equation. Applied Mathematics and Computation 2007; 184(2):895-900.
2. Baykuş Savaşaneril N, Delibaş H., Analytic solution for two-dimensional heat equation for an ellipse region. NTMSCI 4, No. 1, 65-70 (2016)
3. Baykuş Savaşaneril N, Delibaş H., Analytic Solution for The Dirichlet Problem in 2-D Journal of Computational and Theoretical Nanoscience. Vol. 15, 1-5, 2018
4. G. Moretti, Functions of Complex Variable, Prentice-Hall, NJ, 1964.
5. Hacıoğlu Z, Baykuş Savaşaneril N. and Hasan Köse, Solution of Dirichlet problem for a square region in terms of elliptic functions, NTMSCI 3, No. 4, 98-103 (2015)
6. Işık O. R , Sezer M. , Güney Z., Bernstein series solution of linear second-order partial differential equations with mixed conditions, Mathematical Methods in the Applied Sciences, 2013 (wileyonlinelibrary.com)DOI:10.1002/mma.2817
7. Kurul E. and Baykuş Savaşaneril N.,Solution of the two-dimensional heat equation for a rectangular plate. NTMSCI 3, No. 4, 76-82 (2015)
8. Kong W, Wu X. Chebyshev tau matrix method for Poisson-type equations in irregular domain. Journal of Computational and Applied Mathematics 2009; 228(1):158-167.

9. N. Kurt, M. Sezer, A. Çelik, Solution of Dirichlet problem for a rectangular region in terms of elliptic functions, J. Comput. Math. 81 (2004) 1417-1426.
10. N. Kurt, M. Sezer, Solution of Dirichlet problem for a triangle region in terms of elliptic functions, Appl.Math. Comput. 182 (2006) 73-78.
11. N. Kurt, Solution of the two-dimensional heat equation for a square in terms of elliptic functions, Journal of the Franklin Institute, 345(3) 2007 303-317.
12. P. F. Byrd and M.D.Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Lange Maxvelland Springer Ltd. London, New York, 1954
13. Sezer M. Chebyshev polynomial approximation for Dirichlet problem. Journal of Faculty of Science Ege University, Series A 1989; 12(2):69-77.
14. Yüksel G, Işı k O. R and Sezer M. Error analysis of the Chebyshev collocation method for linear second-order partial differential equations, International Journal of Computer Mathematics, 2014 http://dx.doi.org/10.1080/00207160.2014.966099
15. Yüksel G. Chebyshev polynomials solutions of second order linear partial differential equations. Ph.D. Thesis, Muğla University, Muğla, 2011.