Document Type : Research Paper
Authors
- Zahra Kalateh Bojdi ^{1}
- Ataollah Askari Hemmat ^{} ^{2}
- Ali Tavakoli ^{3}
^{1} Department of Mathematics, Faculty of Science and New Technologies, Graduate University of Advanced Technology, Kerman, Iran.
^{2} Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
^{3} Mathematics department, University of Mazandaran, Babolsar, Iran.
Abstract
In this work, the triple convolution of Daubechies wavelet is used to solve the three dimensional (3D) microscale Dual Phase Lag (DPL) problem. Also, numerical solution of 3D time-dependent initial-boundary value problems of a microscopic heat equation is presented. To generate a 3D wavelet we used the triple convolution of a one dimensional wavelet. Using convolution we get a scaling function and a sevenfold 3D wavelet and all of our computations are based on this new set to approximate in 3D spatial. Moreover, approximation in time domain is based on finite difference method. By substitution in the 3D DPL model, the differential equation converts to a linear system of equations and related system is solved directly. We use the Lax-Richtmyer theorem to investigate the consistency, stability and convergence analysis of our method. Numerical results are presented and compared with the analytical solution to show the efficiency of the method.
Keywords
Main Subjects
[2] G. Beylkin and N. Saito, Wavelets, their autocorrelation functions, and multiresolution representation of signals, Expanded abstract in Proceedings ICASSP-92, 4 (1992), pp. 381-384.
[3] C. Canuto, M.Y. Hussaini, A. Quarteroni, and Th.A. Zang, Spectral Methods: Fundamentals in Single Domains, Berlin, Springer, 2006.
[4] C. Canuto, M.Y. Hussaini, A. Quarteroni, and Th.A. Zang, Spectral Methods in Fluid Dynamics, Berlin, Springer Series in Computational Physics, 1988.
[5] G. Chen, Semi-analytical solutions for 2-D modeling of long pulsed laser heating metals with temperature dependent surface absorption, Optik, International Journal for Light and Electron Optics, 2017.
[6] RJ. Chiffell, On the wave behavior and rate effect of thermal and thermo-mechanical waves, M.Sc. Thesis, University of New Mexico, Albuquerque, 1994.
[7] W. Dai, F. Han, and Z. Sun, Accurate Numerical Method for Solving Dual-Phase-Lagging Equation with Temperature Jump Boundary Condition in Nano Heat Conduction, Int. J. Heat Mass Transf., 64 (2013), pp. 966-975.
[8] W. Dai and R. Nassar, A compact finite difference scheme for solving a one-dimensional heat transport equation at the microscale, J. Comput. Appl. Math., 132 (2001), pp. 431-441.
[9] W. Dai and R. Nassar, A compact finite difference scheme for solving a three-dimensional heat transport equation in a thin film, Numer. Methods Partial Differ. Equ., 16 (2000), pp. 441-458.
[10] W. Dai and R. Nassar, A finite difference method for solving the heat transport equation at the microscale, Numer. Methods Partial Differ. Equ., 15 (1999), pp. 697-708.
[11] W. Dai and R. Nassar, A finite difference scheme for solving a three-dimensional heat transport equation in a thin film with microscale thickness, Internat. J. Numer. Methods Engrg., 50 (2001), pp. 1665-1680.
[12] I. Daubechies, Ten Lectures on Wavelets, Soc. for Indtr. Appl. Math., Philadelphia, Number 61, 1992.
[13] J. Fan and L. Wang, Analytical theory of bioheat transport, J. Appl. Phys., 109 (2011).
[14] Z-Y. Guo and Y-S. Xu, Non-Fourier Heat Conduction in IC Chip, ASME J. Electron Packag., 117 (1995), pp. 174-177.
[15] Z. Kalateh Bojdi and A. Askari Hemmat, Wavelet collocation methods for solving the Pennes bioheat transfer equation, Optik, Int. J. Light Electron Optics, 132 (2017), pp. 80-88.
[16] Z. Kargar and H. Saeedi, B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations, Int. J. Wavelets Multiresolut. Inf. Process., 15 (2017), 1750034.
[17] A. Latto, L. Resnikoff, and E. Tenenbaum, The evaluation of connection coefficients of compactly supported wavelets, Proceedings of the French-USA Workshop on Wavelets and Turbulence, 1992, pp. 76-89.
[18] A. Malek, Z. Kalateh Bojdi, and P. Nuri Niled Gobarg, Solving Fully three-Dimensional Micros cal Dual Phase Lag Problem Using Mixed-Collocation, Finite Difference Discretization, Trans. ASME J. Heat Transf., 134 (2012).
[19] A. Malek and SH. Momeni-Masuleh, A Mixed Collocation-Finite Difference Method for 3D Microscopic Heat Transport Problems, J. Comput. Appl. Math., 217 (2008), pp. 137-147.
[20] A. Malek and S.H. Momeni-Masuleh, A Mixed Collocation-Finite Difference Method for 3D Microscopic Heat Transport Problems, J. Comput. Appl. Math., 217 (2008), pp. 137-147.
[21] S. Mallat, Multiresolution approximation and wavelets, Preprint GRASP Lab., Dept. of Computer and Information Science, Univ. of Pennsylvania, 1987.
[22] T.Q. Qui and C.L. Tien, Short-pulse laser heating on metals, Int. J. Heat Mass Transf., 35 (1992), pp. 719-726.
[23] T.Q. Qui and C.L. Tien, Heat transfer mechanisms during short-pulse laser heating on metals, ASME J. Heat Transf., 115 (1993), pp. 835-841.
[24] G.D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods, Third ed., Oxford, Oxford University Press, 1985.
[25] D.Y. Tzou, Macro to Micro Heat Transfer, Washington, Taylor and Francis, 1996.
[26] R. Viskanta and T.L. Bergman, Heat Transfer in Materials Processing, Third Edition, New York, McGraw-Hill Book Company, 1998.
[27] D. Xue, Three-dimensional simulation of the temperature field in high-power double-clad fiber laser, Optik, Int. J. Light Electron Optics, (2011).
[28] J. Zhang and J.J. Zhao, Iterative solution and finite difference approximations to 3D microscale heat transport equation, Math. Comput. Simulation, 57 (2001), pp. 387-404.