A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Jyh-Haw Tang; Chao-Kang Feng

doi:doi:10.11648/j.acm.20221102.11

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A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Jyh-Haw Tang, Chao-Kang Feng

Published in Applied and Computational Mathematics (Volume 11, Issue 2)

Received: 10 March 2022 Accepted: 2 April 2022 Published: 9 April 2022

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Abstract

Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.

Published in	Applied and Computational Mathematics (Volume 11, Issue 2)
DOI	10.11648/j.acm.20221102.11
Page(s)	38-47
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

Keywords

Laplace’s Equation, Fourier Transform, Green’s Function, Similarity Solution

References

[1]	Tang, J. H. (2017). Partial Differential Equations and Boundary Values Problems (in Chinese), Tsang Hai Publishing Company, Taiwan, R.O.C.
[2]	Kevorkian, J. (2010). Partial Differential Equations: Analytical Solution Techniques, 2^nd ed. Springer, New York.
[3]	Myint-U, T., & Debnath, L. (2007) Linear Partial. Differential Equations for Scientists and Engineers. 4^th ed., Springer, New York.
[4]	Gerald, C. F., & Wheatley, P. O. (2004). Applied Numerical Analysis 7^th ed. Pearson Publishing Company, New York.
[5]	Tang, J. H., & Puspasari, A. D. (2021). Numerical Simulation of Local Scour around Three Cylindrical Piles in a Tandem Arrangement. Water MDPI, 13 (24), 3623, 1-19.
[6]	Tang, J. H., Sun, M. K. & Chen, Y. (2013). A Study of Multi-step Overfall Flows by Computational Fluid Dynamics. Applied Mechanics and Materials, 405, 3208-3212.
[7]	Kuo, S. R., Chen, J. T., Lee, J. W., & Chen, Y. W. (2013). Analytical Derivation and Numerical Experiments of Degenerate Scales for Regular N-gon Domains in two-Dimensional Laplace Problems, Applied Mathematics and Computation, 219 (10), 5668-5683.
[8]	Churchill, R. V., & Brown, J. W. (1978). Fourier Series and Boundary Value Problems, 3^rd ed., McGraw-Hill, New York.
[9]	Sedov L. I. (1959). Similarity and Dimensional Methods in Mechanics, Academic Press, New York.
[10]	Bluman G., & Cole J. D. (1974). Similarity Method for Differential Equations, Springer-Verlag, New York.
[11]	Carslaw H. S., & Jaeger J. C. (1963). Operational Methods in Applied Mathematics, 2^nd ed., Dover Publications, New York.
[12]	Churchill, R. V. (1972). Operational Mathematics, 3^rd ed., Mc-Graw-Hill, New York.
[13]	Haberman R. (2004). Elementary Applied Partial Differential Equations 4^th ed., Prentice Hall Inc., New Jersey.
[14]	Stakgold, I., & Hoist, M. J. (2011) Green’s Function and Boundary Value Problems, 3^rd ed., Wiley, New York, United States.
[15]	Feng. C. K. (1989). The General Similarity Solution of Laplace’s Equation with Applications to Boundary Value Problems, Proceeding of 1987 EQUADIFF Conference, Xanthi, Greece, 37-248.
[16]	Feng C. K. (1991). Similarity Analysis of Boundary Value Problems of Poisson’s Equation, International Conference on Differential Equations, Universitat Autonoma de Barcelona, Barcelona, Spain, 26-31.
[17]	Feng C. K., & Lee, C. Y. (2005). The Similarity Analysis of Vibrating Membrane with Its Applications, International Journal of Mechanical Sciences, 47, 961-981.
[18]	Spiegel M. R. (1981). Mathematical Handbook of Formulas and Tables, McGraw-Hill, New York.

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Jyh-Haw Tang, Chao-Kang Feng. (2022). A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Applied and Computational Mathematics, 11(2), 38-47. https://doi.org/10.11648/j.acm.20221102.11

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Jyh-Haw Tang; Chao-Kang Feng. A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Appl. Comput. Math. 2022, 11(2), 38-47. doi: 10.11648/j.acm.20221102.11

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AMA Style

Jyh-Haw Tang, Chao-Kang Feng. A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Appl Comput Math. 2022;11(2):38-47. doi: 10.11648/j.acm.20221102.11

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@article{10.11648/j.acm.20221102.11,
  author = {Jyh-Haw Tang and Chao-Kang Feng},
  title = {A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {38-47},
  doi = {10.11648/j.acm.20221102.11},
  url = {https://doi.org/10.11648/j.acm.20221102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.11},
  abstract = {Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.},
 year = {2022}
}

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TY  - JOUR
T1  - A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions
AU  - Jyh-Haw Tang
AU  - Chao-Kang Feng
Y1  - 2022/04/09
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.11
DO  - 10.11648/j.acm.20221102.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 38
EP  - 47
PB  - Science Publishing Group
SN  - 2328-5613
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AB  - Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.
VL  - 11
IS  - 2
ER  -

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Author Information

Jyh-Haw Tang

Department of Civil Engineering, Chung Yuan Christian University, Tao Yuan City, Taiwan, R.O.C.
Chao-Kang Feng

Department of Aerospace Engineering, TamKang University, New Taipei City, Taiwan, R.O.C.

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Jyh-Haw Tang, Chao-Kang Feng. (2022). A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Applied and Computational Mathematics, 11(2), 38-47. https://doi.org/10.11648/j.acm.20221102.11

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ACS Style

Jyh-Haw Tang; Chao-Kang Feng. A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Appl. Comput. Math. 2022, 11(2), 38-47. doi: 10.11648/j.acm.20221102.11

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AMA Style

Jyh-Haw Tang, Chao-Kang Feng. A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions. Appl Comput Math. 2022;11(2):38-47. doi: 10.11648/j.acm.20221102.11

Copy | Download

@article{10.11648/j.acm.20221102.11,
  author = {Jyh-Haw Tang and Chao-Kang Feng},
  title = {A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {38-47},
  doi = {10.11648/j.acm.20221102.11},
  url = {https://doi.org/10.11648/j.acm.20221102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.11},
  abstract = {Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions
AU  - Jyh-Haw Tang
AU  - Chao-Kang Feng
Y1  - 2022/04/09
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.11
DO  - 10.11648/j.acm.20221102.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 38
EP  - 47
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221102.11
AB  - Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.
VL  - 11
IS  - 2
ER  -

Copy | Download