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Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics

Received: 13 August 2014     Accepted: 11 September 2014     Published: 20 September 2014
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Abstract

Scientists and engineers encounter many kinds of parabolic or hyperbolic distributed dynamics, which are often with inhomogeneous boundary conditions in practice. Boundary inhomogeneity makes the dynamics essentially nonlinear, which prevents the Hilbert space from being applied for modal decomposition and intelligent computation. Thus, this paper systematically deals with this situation via the conversion of the boundary inhomogeneity to a virtual source in conjunction with boundary homogeneity. For such a purpose, the 2D transfer-function is developed based on the Laplace-Galerkin integral transform as the main tool of this conversion. A section of numerical visualization is included to explore the topology of the virtual-source solution. Some interesting findings therein will be addressed.

Published in Applied and Computational Mathematics (Volume 3, Issue 5)
DOI 10.11648/j.acm.20140305.12
Page(s) 197-204
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

Inhomogeneous Boundary Conditions, nD Transfer Function Models, Robin Boundary Conditions, Sturm-Liouville Systems

References
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[2] B.-S. Hong and C.-Y. Chou, “Realization of thermal inertia in frequency domain,” Entropy, vol. 16, pp. 1101-1121, 2014.
[3] B.-S. Hong and C.-Y. Chou, “Energy transfer modelling of active thermoacoustic engines via Lagrangian thermoacoustic dynamics,” Energy Convers. Manage., vol. 84, pp. 73-79, 2014.
[4] B.-S. Hong, V. Yang, and A. Ray, “Robust feedback control of combustion instability with modeling uncertainty,” Combust. Flame, vol. 120, pp. 91-106, 2000.
[5] B.-S. Hong, A. Ray, and V. Yang, “Wide-range robust control of combustion instability,” Combust. Flame, vol. 128, pp. 242-258, 2002.
[6] K. Gustafson and T. Abe, “Gustave Robin: 1855–1897,” Math. Intelligencer, vol. 20, pp. 47–53, 1998.
[7] K. Gustafson and T. Abe, “The third boundary condition—was it Robin’s?,” Math. Intelligencer, vol. 20, no. 1, pp. 63–71, 1998.
[8] A. Romeo and A. A Saharian, “Casimir effect for scalar fields under Robin boundary conditions on plates,” J. Phys. A: Math. Gen,. vol. 35, p. 1297, 2002.
[9] B. Mintz, C. Farina, P. A. Maia Neto, and R. B. Rodrigues, “Particle creation by a moving boundary with a Robin boundary condition,” J. Phys. A: Math. Gen. vol. 39, pp. 11325–11333, 2006.
[10] E. Okada, M. Schweiger, S. R. Arridge, M. Firbank, and D. T. Delpy, “Experimental validation of Monte Carlo and finite-element methods for the estimation of the optical path length in inhomogeneous tissue,” Applied Optics, vol. 35, no. 19, pp. 3362-3371, 1996.
[11] F. Bay, V. Labbe, Y. Favennec, and J. L. Chenot, “A numerical model for induction heating processes coupling electromagnetism and thermomechanics,” Int. J. Numer. Meth. Engng, vol. 58, pp. 839–867, 2003.
[12] B. Jin, “Conjugate gradient method for the Robin inverse problem, associated with the Laplace equation,” Int. J. Numer. Meth. Engng, vol. 71, pp. 433–453, 2007.
[13] X. T. Xiong, X. H. Liu, Y. M. Yan, and H. B. Guo, “A numerical method for identifying heat transfer coefficient,” Appl. Math. Model, vol. 34, pp. 1930–1938, 2010.
[14] E. Majchrzak and M. Paruch, “Identification of electromagnetic field parameters assuring the cancer destruction during hyperthermia treatment,” Inverse Probl. Sci. Eng., vol. 19, no. 1, pp. 45-58, 2011.
[15] B. Jin and X. Lu, “Numerical identification of a Robin coefficient in parabolic problems,” Math. Comp., vol. 81, pp. 1369-1398, 2012.
[16] A. Moosaie, “Axisymmetric non-Fourier temperature field in a hollow sphere,” Arch. Appl. Mech., vol. 79, pp. 679-694 2009.
[17] B. Abdel-Hamid, “Modelling non-Fourier heat conduction with periodic thermal oscillation using the finite integral transform,” Appl. Math. Model. vol. 23, pp. 899–914, 1999.
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    Boe-Shong Hong. (2014). Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics. Applied and Computational Mathematics, 3(5), 197-204. https://doi.org/10.11648/j.acm.20140305.12

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    ACS Style

    Boe-Shong Hong. Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics. Appl. Comput. Math. 2014, 3(5), 197-204. doi: 10.11648/j.acm.20140305.12

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    AMA Style

    Boe-Shong Hong. Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics. Appl Comput Math. 2014;3(5):197-204. doi: 10.11648/j.acm.20140305.12

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  • @article{10.11648/j.acm.20140305.12,
      author = {Boe-Shong Hong},
      title = {Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {5},
      pages = {197-204},
      doi = {10.11648/j.acm.20140305.12},
      url = {https://doi.org/10.11648/j.acm.20140305.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.12},
      abstract = {Scientists and engineers encounter many kinds of parabolic or hyperbolic distributed dynamics, which are often with inhomogeneous boundary conditions in practice. Boundary inhomogeneity makes the dynamics essentially nonlinear, which prevents the Hilbert space from being applied for modal decomposition and intelligent computation. Thus, this paper systematically deals with this situation via the conversion of the boundary inhomogeneity to a virtual source in conjunction with boundary homogeneity. For such a purpose, the 2D transfer-function is developed based on the Laplace-Galerkin integral transform as the main tool of this conversion. A section of numerical visualization is included to explore the topology of the virtual-source solution. Some interesting findings therein will be addressed.},
     year = {2014}
    }
    

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    Y1  - 2014/09/20
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    AB  - Scientists and engineers encounter many kinds of parabolic or hyperbolic distributed dynamics, which are often with inhomogeneous boundary conditions in practice. Boundary inhomogeneity makes the dynamics essentially nonlinear, which prevents the Hilbert space from being applied for modal decomposition and intelligent computation. Thus, this paper systematically deals with this situation via the conversion of the boundary inhomogeneity to a virtual source in conjunction with boundary homogeneity. For such a purpose, the 2D transfer-function is developed based on the Laplace-Galerkin integral transform as the main tool of this conversion. A section of numerical visualization is included to explore the topology of the virtual-source solution. Some interesting findings therein will be addressed.
    VL  - 3
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Author Information
  • Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 62012, Taiwan

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