In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.
| Published in | American Journal of Applied Mathematics (Volume 3, Issue 6) |
| DOI | 10.11648/j.ajam.20150306.20 |
| Page(s) | 305-311 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Initial Condition, Dirichlet Boundary Conditions, Finite Difference Methods, Finite Element Methods, Heat Equation, Instability
| [1] | Alfio Quarteroni, Numerical models for differential problems (2nd edition), Springer - Verlag, Italia, 2014. |
| [2] | Chapra Canal, Numerical methods for engineers (4th edition), The McGraw Hill Companies, 2001. |
| [3] | Erwin Kreyzing, Advanced Engineering Mathematics (9th edition), 2006 John Wiley and Sons, Inc. |
| [4] | G. Evans, J. Blackledge and P. Yardley, Numerical methods for partial differential equations, Springer-Verlag London Limited 2000. |
| [5] | Ioannis P Starroulakis and Stepan A Tersian, Partial Differential equations: An introduction with Mathematica and MAPLE(2nd edition), 2004 by world scientific publishing Co. Plc. Ltd. |
| [6] | J. David Logan, A first course in DEs, 2006 Springer Science + Business Media. Inc. |
| [7] | Lichard L. Burden and J. Douglas Faires, Numerical analysis (9th edition), 2011, 2005, 2001 Brooks/Cole, Cengage Learning, 20 Channel Center Street Boston, MA02210, USA. |
| [8] | Ravi P. Agarwal and Donal O'Regan, Ordinary and Partial DEs, 2006 Springer Science + Business Media, LLC (2009). |
| [9] | Susan C. Brenner, L. Ridgway Scott, The mathematical theory of finite element methods (3rd edition), 2008 Springer Sciences + Business Media, LLC. |
| [10] | V. Dabral, S. Kapoor and S. Dhawan, Numerical Simulation of one dimensional Heat Equation: B-Spline Finite Element Method, V. Dabral et al. / Indian Journal of Computer Science and Engineering (IJCSE), Vol. 2 No. 2 Apr - May 2011. |
APA Style
Benyam Mebrate. (2015). Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. American Journal of Applied Mathematics, 3(6), 305-311. https://doi.org/10.11648/j.ajam.20150306.20
ACS Style
Benyam Mebrate. Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. Am. J. Appl. Math. 2015, 3(6), 305-311. doi: 10.11648/j.ajam.20150306.20
AMA Style
Benyam Mebrate. Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. Am J Appl Math. 2015;3(6):305-311. doi: 10.11648/j.ajam.20150306.20
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20150306.20,
author = {Benyam Mebrate},
title = {Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions},
journal = {American Journal of Applied Mathematics},
volume = {3},
number = {6},
pages = {305-311},
doi = {10.11648/j.ajam.20150306.20},
url = {https://doi.org/10.11648/j.ajam.20150306.20},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.20},
abstract = {In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.},
year = {2015}
}
TY - JOUR T1 - Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions AU - Benyam Mebrate Y1 - 2015/12/25 PY - 2015 N1 - https://doi.org/10.11648/j.ajam.20150306.20 DO - 10.11648/j.ajam.20150306.20 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 305 EP - 311 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20150306.20 AB - In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions. VL - 3 IS - 6 ER -
Information
Email: