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.
| DOI | 10.11648/j.acm.20140301.13 |
| Published in | Applied and Computational Mathematics (Volume 3, Issue 1, February 2014) |
| 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), 2024. Published by Science Publishing Group |
Advection-Diffusion Problem, Variable Diffusion Constant, Integral Representation Method, Primary Space-Differential Operator, Generalized Fundamental Solution, Generalized Integral Representation Method Component
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APA Style
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
ACS 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
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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Download@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}
}
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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