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Theory and Application of the Generalized Integral Representation Method (GIRM) in Advection Diffusion Problem

Received: 9 July 2014     Accepted: 29 July 2014     Published: 10 August 2014
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Abstract

The integral representation is developed for linear initial and boundary value problems. The fundamental solution is defined by the linear differential equation with constant coefficients and plays a key role in obtaining the integral representation. This becomes a very strong constraint in developing the theory to nonlinear problems. In the present paper, an innovative generalization of the integral representation or generalized integral representation is proposed. The numerical examples are given to verify the theory.

Published in Applied and Computational Mathematics (Volume 3, Issue 4)
DOI 10.11648/j.acm.20140304.15
Page(s) 137-149
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, Reciprocity, Integral Representation, Fundamental Solution, Generalization

References
[1] Wu J.C., Thompson J.F., “Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulations”, Computers & Fluids, (1973), 1, pp. 197-215.
[2] S. J. Uhlman, “An integral equation formulation of the equations of motion of an incompressible fluid”, NUWC-NPT Technical Report 10,086, 15 July, (1992).
[3] H. Isshik, S. Nagata, Y. Imai, “Solution of Viscous Flow around a Circular Cylinder by a New Integral Representation Method (NIRM)”, AJET, 2, 2, (2014), pp. 60-82. file:///C:/Users/l/Downloads/983-5001-1-PB%20(1).pdf
[4] H. Isshik, S. Nagata, Y. Imai, “Solution of a diffusion problem in a non-homogeneous flow and diffusion field by the integral representation method (IRM)”, Applied Mathematics and Computation, 3(1), (2014), pp. 15-26. http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.13.pdf
[5] H. Isshiki, “Improvement of Stability and Accuracy of Time-Evolution Equation by Implicit Integration”, Asian Journal of Engineering and Technology (AJET), Vol. 2, No. 2 (2014), pp. 1339–160. file:///C:/Users/l/Downloads/1205-5161-1-PB.pdf
[6] H. Isshiki, A method for Reduction of Spurious or Numerical Oscillations in Integration of Unsteady Boundary Value Problem, AJET, 2, 3, (2014), pp. 190-202. file:///C:/Users/l/Downloads/1360-5725-2-PB%20(2).pdf
[7] L. B. Lucy, “A numerical approach to the testing of the fission hypothesis”, The Astronomical Journal, vol. 82, no. 12 (1977), pp. 1013-1024. http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977AJ.....82.1013L&defaultprint=YES&filetype=.pdf.
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  • APA Style

    H. Isshiki. (2014). Theory and Application of the Generalized Integral Representation Method (GIRM) in Advection Diffusion Problem. Applied and Computational Mathematics, 3(4), 137-149. https://doi.org/10.11648/j.acm.20140304.15

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    ACS Style

    H. Isshiki. Theory and Application of the Generalized Integral Representation Method (GIRM) in Advection Diffusion Problem. Appl. Comput. Math. 2014, 3(4), 137-149. doi: 10.11648/j.acm.20140304.15

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    AMA Style

    H. Isshiki. Theory and Application of the Generalized Integral Representation Method (GIRM) in Advection Diffusion Problem. Appl Comput Math. 2014;3(4):137-149. doi: 10.11648/j.acm.20140304.15

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  • @article{10.11648/j.acm.20140304.15,
      author = {H. Isshiki},
      title = {Theory and Application of the Generalized Integral Representation Method (GIRM) in Advection Diffusion Problem},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {4},
      pages = {137-149},
      doi = {10.11648/j.acm.20140304.15},
      url = {https://doi.org/10.11648/j.acm.20140304.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.15},
      abstract = {The integral representation is developed for linear initial and boundary value problems. The fundamental solution is defined by the linear differential equation with constant coefficients and plays a key role in obtaining the integral representation. This becomes a very strong constraint in developing the theory to nonlinear problems. In the present paper, an innovative generalization of the integral representation or generalized integral representation is proposed. The numerical examples are given to verify the theory.},
     year = {2014}
    }
    

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    AB  - The integral representation is developed for linear initial and boundary value problems. The fundamental solution is defined by the linear differential equation with constant coefficients and plays a key role in obtaining the integral representation. This becomes a very strong constraint in developing the theory to nonlinear problems. In the present paper, an innovative generalization of the integral representation or generalized integral representation is proposed. The numerical examples are given to verify the theory.
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Author Information
  • IMA (Institute of Mathematical Analysia), Osaka, Japan

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