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Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)

Received: 19 March 2015     Accepted: 23 March 2015     Published: 8 April 2015
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Abstract

In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial and Boundary Value Problem for advective diffusion. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The corresponding computer codes implemented in most popular computational languages are also given.

Published in Applied and Computational Mathematics (Volume 4, Issue 3-1)

This article belongs to the Special Issue Integral Representation Method and its Generalization

DOI 10.11648/j.acm.s.2015040301.15
Page(s) 59-77
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), 2015. Published by Science Publishing Group

Keywords

Initial and Boundary Value Problem (IBVP), Generalized Fundamental Solution, Generalized Integral Representation Method (GIRM), Implementation of GIRM, Computer Codes

References
[1] H. Isshiki, “From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM),” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggroup.com/ journal/archive.aspx?journalid=147&issueid=-1
[2] H. Isshiki, T. Takiya, and H. Niizato, “Application of Generalized Integral representation (GIRM) Method to Fluid Dynamic Motion of Gas or Particles in Cosmic Space Driven by Gravitational Force,” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggroup.com/journal/archive.aspx?journalid=147&issueid=-1
[3] H. Isshiki, “Effects of Generalized Fundamental Solution (GFS) on Generalized Integral Representation Method (GIRM),” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggro up .com/journal/archive.aspx?journalid=147&issueid=-1
[4] H. Isshiki, “Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation,” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggroup.com/ journal/archive.aspx?journalid=147&issueid=-1
Cite This Article
  • APA Style

    Hideyuki Niizato, Gantulga Tsedendorj, Hiroshi Isshiki. (2015). Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs). Applied and Computational Mathematics, 4(3-1), 59-77. https://doi.org/10.11648/j.acm.s.2015040301.15

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    ACS Style

    Hideyuki Niizato; Gantulga Tsedendorj; Hiroshi Isshiki. Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs). Appl. Comput. Math. 2015, 4(3-1), 59-77. doi: 10.11648/j.acm.s.2015040301.15

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    AMA Style

    Hideyuki Niizato, Gantulga Tsedendorj, Hiroshi Isshiki. Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs). Appl Comput Math. 2015;4(3-1):59-77. doi: 10.11648/j.acm.s.2015040301.15

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  • @article{10.11648/j.acm.s.2015040301.15,
      author = {Hideyuki Niizato and Gantulga Tsedendorj and Hiroshi Isshiki},
      title = {Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {3-1},
      pages = {59-77},
      doi = {10.11648/j.acm.s.2015040301.15},
      url = {https://doi.org/10.11648/j.acm.s.2015040301.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2015040301.15},
      abstract = {In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial and Boundary Value Problem for advective diffusion. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The corresponding computer codes implemented in most popular computational languages are also given.},
     year = {2015}
    }
    

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  • TY  - JOUR
    T1  - Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)
    AU  - Hideyuki Niizato
    AU  - Gantulga Tsedendorj
    AU  - Hiroshi Isshiki
    Y1  - 2015/04/08
    PY  - 2015
    N1  - https://doi.org/10.11648/j.acm.s.2015040301.15
    DO  - 10.11648/j.acm.s.2015040301.15
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 59
    EP  - 77
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.s.2015040301.15
    AB  - In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial and Boundary Value Problem for advective diffusion. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The corresponding computer codes implemented in most popular computational languages are also given.
    VL  - 4
    IS  - 3-1
    ER  - 

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Author Information
  • Hitachi Zosen Corporation, Osaka, Japan

  • Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia

  • IMA, Institute of Mathematical Analysis, Osaka, Japan

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