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Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM)

Received: 10 January 2014     Published: 20 February 2014
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Abstract

Integral representations are derived from a differential-type boundary value problem using a fundamental solution. A set of integral representations is equivalent to a set of differential equations. If the boundary conditions are substituted into the integral representations, the integral equations are obtained, and the unknown variables are determined by solving the integral equations. In other words, an integral-type boundary value problem is derived from the integral representations. An effective and flexible finite element algorithm is easily obtained from the integral-type boundary value problem. In the present paper, integral representations are obtained for the diffusion of a material or heat in the sea, where the convective velocity and diffusion constant change in space and time. A new numerical solution of an advection-diffusion equation is proposed based integral representations using the fundamental solution of the primary space-differential operator, and the numerical results are shown. An innovative generalization of the integral representation method: generalized integral representation method is also proposed. The numerical examples are given to verify the theory.

Published in Applied and Computational Mathematics (Volume 3, Issue 1)
DOI 10.11648/j.acm.20140301.13
Page(s) 15-26
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), 2014. Published by Science Publishing Group

Keywords

Advection-Diffusion Problem, Variable Diffusion Constant, Integral Representation Method, Primary Space-Differential Operator, Generalized Fundamental Solution, Generalized Integral Representation Method Component

References
[1] C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, 4.3 Coupled boundary element - Finite difference methods, Boundary Element Techniques, Theory and Applications in Engineering, Springer-Verlag (1984).
[2] B. Nayroles, G. Touzot, P. Villon, Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mechanics 10 (1992) 307-318.
[3] T. Belytschko, Y.Y. Lu, L. Gu, Element-free Galarkin methods, International Journal for Numerical Methods in Engineering 37 (1994) 229-256.
[4] G. Yagawa, Y. Yamda, Free mesh method: A new meshless finite element method, Computational Mechanics 18 (1996) 383-386.
[5] L.B. Lucy, A numerical approach to the testing of the fission hypotheis, The Astronomical Jounal 82 (12) (1977) 1013-1024.
[6] G.R. Liu, M.B. Liu, Smoothed Particle Hydrodyndmics–a meshfree particle method. World Scientific; ISBN 981-238-456-1 (2003).
[7] H. Isshiki, Discrete differential operators on irregular nodes (DDIN), International Journal for Numerical Methods in Engineering 88 (12) (2011) 1323-1343.
[8] H. Isshiki, Random Collocation Method (RCM), IMECE2010-39054, Vancouver, Canada (2010).
[9] J.S. Uhlman, An integral equation formulation of the equations of motion of an incompressible fluid, NUWC-NPT Technical Report 10,086 15 July (1992).
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    Hiroshi Isshiki, Shuichi Nagata, Yasutaka Imai. (2014). Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM). Applied and Computational Mathematics, 3(1), 15-26. https://doi.org/10.11648/j.acm.20140301.13

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    Hiroshi Isshiki; Shuichi Nagata; Yasutaka Imai. Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM). Appl. Comput. Math. 2014, 3(1), 15-26. doi: 10.11648/j.acm.20140301.13

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    AMA Style

    Hiroshi Isshiki, Shuichi Nagata, Yasutaka Imai. Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM). Appl Comput Math. 2014;3(1):15-26. doi: 10.11648/j.acm.20140301.13

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  • @article{10.11648/j.acm.20140301.13,
      author = {Hiroshi Isshiki and Shuichi Nagata and Yasutaka Imai},
      title = {Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM)},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {1},
      pages = {15-26},
      doi = {10.11648/j.acm.20140301.13},
      url = {https://doi.org/10.11648/j.acm.20140301.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.13},
      abstract = {Integral representations are derived from a differential-type boundary value problem using a fundamental solution. A set of integral representations is equivalent to a set of differential equations. If the boundary conditions are substituted into the integral representations, the integral equations are obtained, and the unknown variables are determined by solving the integral equations. In other words, an integral-type boundary value problem is derived from the integral representations. An effective and flexible finite element algorithm is easily obtained from the integral-type boundary value problem. In the present paper, integral representations are obtained for the diffusion of a material or heat in the sea, where the convective velocity and diffusion constant change in space and time. A new numerical solution of an advection-diffusion equation is proposed based integral representations using the fundamental solution of the primary space-differential operator, and the numerical results are shown. An innovative generalization of the integral representation method: generalized integral representation method is also proposed. The numerical examples are given to verify the theory.},
     year = {2014}
    }
    

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  • TY  - JOUR
    T1  - Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM)
    AU  - Hiroshi Isshiki
    AU  - Shuichi Nagata
    AU  - Yasutaka Imai
    Y1  - 2014/02/20
    PY  - 2014
    N1  - https://doi.org/10.11648/j.acm.20140301.13
    DO  - 10.11648/j.acm.20140301.13
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 15
    EP  - 26
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20140301.13
    AB  - Integral representations are derived from a differential-type boundary value problem using a fundamental solution. A set of integral representations is equivalent to a set of differential equations. If the boundary conditions are substituted into the integral representations, the integral equations are obtained, and the unknown variables are determined by solving the integral equations. In other words, an integral-type boundary value problem is derived from the integral representations. An effective and flexible finite element algorithm is easily obtained from the integral-type boundary value problem. In the present paper, integral representations are obtained for the diffusion of a material or heat in the sea, where the convective velocity and diffusion constant change in space and time. A new numerical solution of an advection-diffusion equation is proposed based integral representations using the fundamental solution of the primary space-differential operator, and the numerical results are shown. An innovative generalization of the integral representation method: generalized integral representation method is also proposed. The numerical examples are given to verify the theory.
    VL  - 3
    IS  - 1
    ER  - 

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Author Information
  • Institute of Ocean Energy, Saga University, Saga, Japan

  • Institute of Ocean Energy, Saga University, Saga, Japan

  • Institute of Ocean Energy, Saga University, Saga, Japan

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