Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics
Applied and Computational Mathematics
Volume 3, Issue 5, October 2014, Pages: 197-204
Received: Aug. 13, 2014;
Accepted: Sep. 11, 2014;
Published: Sep. 20, 2014
Views 2585 Downloads 146
Boe-Shong Hong, Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 62012, Taiwan
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.
Realization of Inhomogeneous Boundary Conditions as Virtual Sources in Parabolic and Hyperbolic Dynamics, Applied and Computational Mathematics.
Vol. 3, No. 5,
2014, pp. 197-204.
B.-S. Hong, “Construction of 2D isomorphism for 2D -control of Sturm-Liouville Systems,” Asian J. Control, vol. 12, no. 2, pp. 187-199, 2010.
B.-S. Hong and C.-Y. Chou, “Realization of thermal inertia in frequency domain,” Entropy, vol. 16, pp. 1101-1121, 2014.
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.
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.
B.-S. Hong, A. Ray, and V. Yang, “Wide-range robust control of combustion instability,” Combust. Flame, vol. 128, pp. 242-258, 2002.
K. Gustafson and T. Abe, “Gustave Robin: 1855–1897,” Math. Intelligencer, vol. 20, pp. 47–53, 1998.
K. Gustafson and T. Abe, “The third boundary condition—was it Robin’s?,” Math. Intelligencer, vol. 20, no. 1, pp. 63–71, 1998.
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.
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.
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.
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.
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.
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.
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.
B. Jin and X. Lu, “Numerical identification of a Robin coefficient in parabolic problems,” Math. Comp., vol. 81, pp. 1369-1398, 2012.
A. Moosaie, “Axisymmetric non-Fourier temperature field in a hollow sphere,” Arch. Appl. Mech., vol. 79, pp. 679-694 2009.
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.
G. Heidarinejad, R. Shirmohammadi, and M. Maerefat, “Heat wave phenomena in solids subjected to time dependent surface heat flux,” Heat Mass Transf., vol. 44, pp. 381–392, 2008.
N. Young, An Introduction to Hilbert Space. Cambridge University Press, 1988.