Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation
Mathematics and Computer Science
Volume 5, Issue 4, July 2020, Pages: 76-85
Received: Mar. 10, 2020;
Accepted: Apr. 7, 2020;
Published: Oct. 12, 2020
Views 198 Downloads 84
Kedir Aliyi Koroche, Department of Mathematics, College of Natural Sciences, Ambo University, Ambo, Ethiopia
In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E∞) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.
Kedir Aliyi Koroche,
Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation, Mathematics and Computer Science.
Vol. 5, No. 4,
2020, pp. 76-85.
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Benyam Mabrate (2015). Numerical solution of a one-dimensional Heat Equation with Dirichlet boundary conditions. American Journal of Applied Mathematics, 3 (4): 305-311.
Bernatz, R. (2010). Fourier series and numerical methods for partial differential equations. John Wiley and Sons.
Cheninguael, A. (2014, March). Numerical Method for the Heat Equation with Dirichlet and Neumann Conditions. In Proceedings of the International Multiconference of Engineering and computer scientists (vol. 1).
Dabral, V., Kapoor, S. and Dhawan, S. (Apr-May 2011). Numerical Simulation of one dimensional Heat Equation: B-Spline Finite Element Method, Indian Jour nal of Computer Science and Engineering. 2 (2), 222-235.
Fasshauer, G. (2007). Meshfree Application Method with Matlab. Interdisciplinary Mathematical Sciences.
Franke, R. (1982). Scattered data interpolation: tests of some methods. Mathematics of computation, 38 (157): 181-200.
Francis B. Hildebrand (1987). Introduction to numerical analysis, Second-edition, Dover Publications-Inc., Canada.
Hooshmandasl M. R Haidari, M., and MaalekGhaini F. M. (2012). Numerical solution of one dimensional heat equation by using the Chebyshev Wavelets method. JACM an open-access journal, 1 (6), 1-7.
Kalyanil, P., and Rao (2013). Numerical solution of heat equation through double Interpolation. IOSR Journal. of math. 6 (6): 58-62.
Li, J. R., and Greengard, L. (2007). On the numerical solution of the heat equation I: Fast solvers in free space. Journal of Computational Physics, 226 (2): 1891-1901.
Muluneh Dingeta, Gemechis File and Tesfaye Aga (2018) Numerical Solution of Second Numerical Solution of Second-Order One Dimensional Linear Hyperbolic Telegraph Equation Ethiop. J. Educ. & Sc. Vol. 14 No 1.
Rashidinia J. Esfahani F. and Jamalzadeh S., (2013), B-spline Collocation Approach for Solution of Klein-Gordon Equation. International Journal of Mathematical Modeling and Computations 3, 25-33.
Ray S, S., (2016) Numerical analysis with algorithms and programming, Taylor & Francis Group, LLC, USA.
Schaback, R. (1995). Error estimates and condition numbers for radial basis func tion interpolation. Advances in Computational Mathematics, 3 (3), 251-264.
Shokofeh S. and Rashidinia J. (2016), Numerical solution of the hyperbolic telegraph the equation by cubic B-spline collocation method. Applied Mathematics and Computation 281, 28–38.
Sastry, S. S. (2006). Introductory method of numerical analysis, Fourth-edition, Asoke Ghash, prentice Hall of India.
Schiesser, W. E., and Griffiths, G. W. (2009). A compendium of partial different tial equation models: method of lines analysis with Matlab. Cambridge University Press.
Tatari, M. and Dehghan, M., (2010). A method for solving partial differential equations via radial basis functions: Application to the heat equation Engineering Analysis with Boundary element 34 (3), 206-212.
Terefe Asrat, Gemechis File and Tesfaye Aga (2016), Fourth-order stable central difference method for selfadjoint singular perturbation problems, Ethiop. J. Sci. & Technol. 9 53-68.
Hikmet¸ ag˘lar, Mehmet O. zer, Nazan¸ ag˘lar (2008) The numerical solution of the one-dimensional heat equation by using third degree B-spline functions.