On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)

Mohammad Roknujjaman; Mohammad Asaduzzaman

doi:doi:10.11648/j.ajam.20180601.11

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On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)

Mohammad Roknujjaman, Mohammad Asaduzzaman

Published in American Journal of Applied Mathematics (Volume 6, Issue 1)

Received: 28 December 2017 Accepted: 16 January 2018 Published: 12 February 2018

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.ajam.20180601.11
Page(s)	1-7
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), 2018. Published by Science Publishing Group

Keywords

Dirichlet’s Boundary Condition, Laplace Equation, MOL, PDE, CNM

References

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[3]	A. G. Bratsos, “The solution of the Boussinesq equation using the method of lines”, Comput. Methods Appl. Mech. Eng. 157, 33–44 (1998).
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[7]	G. Hall and J. M. Watt, “Modern Numerical Methods for Ordinary Differential Equations, Clarendon”, Oxford (1976).
[8]	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).
[9]	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).
[10]	T. Koto, “Method of lines approximations of delay differential equations”, Comput. Math. Appl. 48, 45–59 (2004).
[11]	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).
[12]	W. E. Schiesser, “The Numerical Method of Lines”, Academic Press, San Diego.
[13]	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).
[14]	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).
[15]	A. A. Sharaf and H. O. Bakodah,“A Good Spatial Discretization in the Method of Lines”, Applied Mathematics and Computation” (2005).
[16]	Z. A. Vlasova, “A Numerical Realization of the Method of Reduction to Ordinary Differential Equations”, Sibirsk. Math. Za. 4, 475-479 (1963).
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[18]	G. D. Smith, “Numerical Solution of Partial Differential Equations (Crank-Nicholson Methods)”, 3rd Edition, Oxford University Press, Oxford (1985).
[19]	S. G. Mikhlin, and K. L. Smolitskii, “Approximate Methods for Solving Differential and Integral Equations”, Moscow 1965, pp 329-335 (1965).
[20]	H. O. Kreiss and J. Lorenz, “Initial-Boundary Value Problems and the Navier-Stokes Equations”, SIAM, Philadelphia (2004).

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APA Style

Mohammad Roknujjaman, Mohammad Asaduzzaman. (2018). 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, 6(1), 1-7. https://doi.org/10.11648/j.ajam.20180601.11

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ACS Style

Mohammad Roknujjaman; Mohammad Asaduzzaman. On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM). Am. J. Appl. Math. 2018, 6(1), 1-7. doi: 10.11648/j.ajam.20180601.11

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AMA Style

Mohammad Roknujjaman, Mohammad Asaduzzaman. On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM). Am J Appl Math. 2018;6(1):1-7. doi: 10.11648/j.ajam.20180601.11

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@article{10.11648/j.ajam.20180601.11,
  author = {Mohammad Roknujjaman and Mohammad Asaduzzaman},
  title = {On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajam.20180601.11},
  url = {https://doi.org/10.11648/j.ajam.20180601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180601.11},
  abstract = {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.},
 year = {2018}
}

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TY  - JOUR
T1  - On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)
AU  - Mohammad Roknujjaman
AU  - Mohammad Asaduzzaman
Y1  - 2018/02/12
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180601.11
DO  - 10.11648/j.ajam.20180601.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
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PB  - Science Publishing Group
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AB  - 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.
VL  - 6
IS  - 1
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Author Information

Mohammad Roknujjaman

Department of Computer Science and Engineering, Northern College Bangladesh, Dhaka, Bangladesh
Mohammad Asaduzzaman

Department of Mathematics, Islamic University (IU), Kushtia, Bangladesh

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APA Style

Mohammad Roknujjaman, Mohammad Asaduzzaman. (2018). 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, 6(1), 1-7. https://doi.org/10.11648/j.ajam.20180601.11

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ACS Style

Mohammad Roknujjaman; Mohammad Asaduzzaman. On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM). Am. J. Appl. Math. 2018, 6(1), 1-7. doi: 10.11648/j.ajam.20180601.11

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AMA Style

Mohammad Roknujjaman, Mohammad Asaduzzaman. On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM). Am J Appl Math. 2018;6(1):1-7. doi: 10.11648/j.ajam.20180601.11

Copy | Download

@article{10.11648/j.ajam.20180601.11,
  author = {Mohammad Roknujjaman and Mohammad Asaduzzaman},
  title = {On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajam.20180601.11},
  url = {https://doi.org/10.11648/j.ajam.20180601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180601.11},
  abstract = {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.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)
AU  - Mohammad Roknujjaman
AU  - Mohammad Asaduzzaman
Y1  - 2018/02/12
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180601.11
DO  - 10.11648/j.ajam.20180601.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180601.11
AB  - 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.
VL  - 6
IS  - 1
ER  -

Copy | Download