On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)
American Journal of Applied Mathematics
Volume 6, Issue 1, February 2018, Pages: 1-7
Received: Dec. 28, 2017;
Accepted: Jan. 16, 2018;
Published: Feb. 12, 2018
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Mohammad Roknujjaman, Department of Computer Science and Engineering, Northern College Bangladesh, Dhaka, Bangladesh
Mohammad Asaduzzaman, Department of Mathematics, Islamic University (IU), Kushtia, Bangladesh
In this paper we used the method of lines (MOL) as a solution procedure for solving partial differential equation (PDE). The range of applications of the MOL has increased dramatically in the last few years; nevertheless, there is no introductory to initiate a beginner to the method. This Paper illustrates the application of the MOL using Crank-Nicholson method (CNM) for numerical solution of PDE together with initial condition and Dirichlet’s Boundary Condition. The implementation of this solutions is done using Microsoft office excel worksheet or spreadsheet, Matlab programming language. Finally, here we analysis the particular solution and numerical solution of Laplace equation obtained by MOL along with CNM.
On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM), American Journal of Applied Mathematics.
Vol. 6, No. 1,
2018, pp. 1-7.
Copyright © 2018 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.
I. Ayranci and N. Selçuk, “MOL solution of DOM for transient radiative transfer in 3-D scattering media”, J. Quant. Spectrosc. Radiat. Transfer 84, 409–422 (2004).
A. G. Bratsos, “The solution of the two-dimensional sine-Gordon equation using the method of lines”, J. Comput. Appl. Math. 206, 251–277 (2007).
A. G. Bratsos, “The solution of the Boussinesq equation using the method of lines”, Comput. Methods Appl. Mech. Eng. 157, 33–44 (1998).
A. G. Bratsos and E. H. Twizell, “The solution of the sine-Gordon equation using the method of lines”, Int. J. Comput. Math. 61, 271–292 (1996).
V. D. Egorov, “The mehtod of lines for condensation kinetics problem solving”, J. Aerosol Sci. 30 (Suppl. I), S247–S248 (1999).
A. Erdem and S. Pamuk, “The method of lines for the numerical solution of a mathematical model for capillary formation: the role of tumor angiogenic factor in the extra-cellular matrix”, Appl. Math. Comput. 186, 891–897 (2007).
G. Hall and J. M. Watt, “Modern Numerical Methods for Ordinary Differential Equations, Clarendon”, Oxford (1976).
S. Hamdi, W. H. Enright, Y. Ouellet and W. E. Schiesser, “Method of lines solutions of the extended Boussinesq equations”, J. Comput. Appl. Math. 183, 327–342 (2005).
H. Han and Z. Huang, “The direct method of lines for the numerical solution of interface problem”, Comput. Methods Appl. Mech. Eng. 171, 61–75 (1999).
T. Koto, “Method of lines approximations of delay differential equations”, Comput. Math. Appl. 48, 45–59 (2004).
S. Pamuk and A. Erdem, “The method of lines for the numerical solution of a mathematical model for capillary formation: the role of endothelial cells in the capillary”, Appl. Math. Comput. 186, 831–835 (2007).
W. E. Schiesser, “The Numerical Method of Lines”, Academic Press, San Diego.
D. J. Jones et al, “On the numerical solution of elliptic partial differential equations by the method of lines”, J. Comput. Phys., 9, 496–527 (1972).
J. G. Ma and Z. Chen, “Application of the method of lines to the Laplace equation”, Micro. Opt. Tech. L ett., 14 (6), 330–333 (1997).
A. A. Sharaf and H. O. Bakodah,“A Good Spatial Discretization in the Method of Lines”, Applied Mathematics and Computation” (2005).
Z. A. Vlasova, “A Numerical Realization of the Method of Reduction to Ordinary Differential Equations”, Sibirsk. Math. Za. 4, 475-479 (1963).
W. E. Schiesser, and G. W. Griffths, “A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab”, Cambridge University Press (2009).
G. D. Smith, “Numerical Solution of Partial Differential Equations (Crank-Nicholson Methods)”, 3rd Edition, Oxford University Press, Oxford (1985).
S. G. Mikhlin, and K. L. Smolitskii, “Approximate Methods for Solving Differential and Integral Equations”, Moscow 1965, pp 329-335 (1965).
H. O. Kreiss and J. Lorenz, “Initial-Boundary Value Problems and the Navier-Stokes Equations”, SIAM, Philadelphia (2004).