臺灣博碩士論文加值系統

欲解決現實中所遇到物理問題時，首先需要把物理系統化成數學模型來解，而大多數自然界的現象都含著非線性物理行為，因此化成數學模型時會有著非線性項，非線性項會使得求解的難度上增加。
本研究旨在利用Laplace轉換法配合 Adomian 分解法(簡稱LADM法)來求解非線性熱傳系統中的近似解析解，討論鰭片、熱處理加工中的慢速移動板，於對流、輻射、以及隨著溫度變化的熱傳導係數、隨著溫度變化的表面放射率，比對各項參數來做討論，並把結果與其他文獻做比較，發現LADM法前五項的總和已相當吻合。
研究結果顯示當無因次化參數Nc(熱對流係數相對於熱傳導數的比值)增加、Nr(熱輻射率相對於熱傳導數的比值)增加、B(表面放射率對於溫度的斜率)增加均會使得板子散熱更好，而A(熱傳導係數對於溫度的斜率)增加表示熱傳遞的更快，因此經過一固定長度的鰭片或板子溫度降則是會更少，在Pe(Peclet number)方面，若假設熱傳過一固定長度且熱擴散係數不變，表示移動越快會使得最終溫度增加。假設熱對流系數呈現幂級次變化，次數越大對流整體效益越小。次數越大數學系統所產生的非線性項也越多。
本研究目的在對於往後鰭片、加工板的材料選擇、對流流體選擇尺度上有進一步認知，並且說明了LADM可以有效、快速解決非線性物理問題，在使用LADM法時必須注意其疊代的收斂情形、初始項的選擇考量、函數Laplace轉換是否存在，若無法收斂時必須以近似法來協助修正。

When we try to solve physical problems ,we usually build math models to approach our problems. Nonlinear terms are very common both in physical problems and math models, but they will complicate the solving process.
In this paper , we use LADM method to solve nonlinear heat transfer problems. We have two cases , nonlinear fin system and nonlinear continuously moving plates system. Some parameters like Convection, radiation, slope of the thermal conductivity-temperature curve , slope of the surface emissivity-temperature curve are discussed.
We found when dimensionless numbers as following increase：Nc、Nr 、B(which presents the conventional intensity to conductional intensity、radiative intensity to conductional intensity、surface emission , respectively)will speed up heat transfer in fin or plate. Dimensionless number A(which presents slope of the thermal conductivity-temperature curve)increases will make heat transfer more faster in fin or plate. Dimensionless number Pe(which presents peclet number) increases (if we have a constant fin length or plate length and constant thermal diffusivity )will make final temperature higher. Assuming a power law variation (decided by parameter m ) of the convection coefficient . Nonlinear terms can also be increased by parameter m .
In conclusion, LADM is an effective way to solve nonlinear system. Following this paper , we can know more in material or fluid selecting.

中文摘要I
英文延伸摘要III
誌謝XII
目錄XIII
圖目錄XVI
表目錄XVII
符號說明XVIII

第一章 緒論1
1-1前言1
1-2文獻回顧2
1-2-1 非線性物理系統之發展2
1-2-2 非線性熱傳系統3
1-2-3 Adomian分解法4
1-2-4 Laplace Adomian分解法5
1-3本文架構7
第二章 Laplace Adomian混合分解法8
2-1 Adomian分解法 8
2-2 Adomian多項式 12
2-2-1非線性(Nonlinear)多項式12
2-2-2非線性微分(Nonlinear derivative)多項式14
2-2-3指數(Exponential)函數多項式17
2-2-4對數(Logarithmic)函數多項式18
2-2-5三角(Trigonometric)函數多項式20
2-3 修正Adomian分解法23
2-3-1修正ADM（一)24
2-3-2修正ADM（二)26
2-4 Laplace Adomian分解法 28
2-4-1運算法則29
2-4-2討論31
第三章 鰭片非線性熱傳分析36
3-1物理模型36
3-2 Laplace Adomian分解法解題步驟39
3-3 結果47
3-4 討論57
第四章 熱處理加工非線性熱傳分析62
4-1物理模型63
4-2 Laplace Adomian分解法解題步驟66
4-3 結果71
4-4 討論74
第五章 總結與未來展望78
5-1 總結78
5-2 未來展望79
參考文獻80

A.M.Wazwaz , 1998, “A comparison between Adomian decomposition method and Taylor series method in the series solutions, Applied Mathematics and Computation, Vol. 97, pp.37-44.
A.M.Wazwaz , 1999, “Analytical approximations and Padé approximants for Volterra’s population model, Applied Mathematics and Computation, Vol. 100, pp.13-25.
A.M.Wazwaz , 1999, “A reliable modification of Adomian decomposition method, Applied Mathematics and Computation, Vol. 102, pp. 77-86.
A.M.Wazwaz, 1999, “The modified decomposition method and Padé approximants for solving the Thomas–Fermi equation, Applied Mathematics and Computation, Vol. 105, pp.11-19.
A.M.Wazwaz , 2000, “A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation, Vol. 111, pp.53-69.
A.M.Wazwaz, 2001, “A reliable algorithm for solving boundary layer problems for higher-order integro-differential equations, Applied Mathematics and Computation, Vol. 118, pp. 327-342.
A.M.Wazwaz , 2010, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volteera integro-differential equations “Applied Mathematics and Computation , vol.216,pp.1304-1309
A.Aziz, F.Khani, 2010 ,Convection-radiation from a continuously moving fin of variable thermal conductivity Journal of the Franklin Institute , Vol. 348, pp. 640-651
A.Aziz, R.J.Lopez, 2011 , “Convection-radiation from a continuously moving variable thermal conductivity sheet or rod undergoing thermal processing, International Journal of Thermal Sciences,vol.50,pp.1523-1531
A.Aziz, M.Torabi, K.Zhang , 2013 , “Convective–radiative radial fins with convective base heating and convective–radiative tip coolinghomogeneous and functionally graded materials Energy Conversion and Management,vol.75,pp.366-376
A.S. Arife, S.T. Korashe, A. Yildirim, 2011 ,Laplace adomian decomposition method for solving a model chronic myelogenouus leukemia CML and T cell interaction, World Applied Sciences Journal 13(4)
B. Van der Pol, J. Van der Mark, 1927, “Frequency demultiplication, Nature, Vol. 120, No. 3019, pp. 363-364.
C. Arslanturk , 2009, “Correlation equations for optimum design of annular fins with temperature dependent thermal conductivityHeat Mass Transfer vol.45,pp.519-525
F. Khani , M. A Raji , H.H. Nejad , 2009 , “Analytical solutions and efficiency of the nonlinear fin problem with temperature dependent thermal conductivity and heat transfer coefficient ,Communi Nonlinear Sci Simulation , vol.14,pp . 3327 – 3338
F. Incropera ：Introduction to Heat Transfer 5th edition
G. Adomian , 1995 , “Delayed Nonlinear Dynamical Systems, Mathematical and Computer Modelling, vol.22,No3,pp. 77-79


G.Adomian, 1996, “Solution of the coupled nonlinear partial differential equations by decomposition, Computers ＆ Mathematics with Applications, Vol. 31, No. 6, pp. 117-120.
G.Adomian, 1996, “The fifth-order Korteweg-de Vries equation, International Journal of Mathematics and Mathematical Sciences, Vol. 19, No. 2, pp. 415.
G.Adomian, 1997, “On the dynamics of a reaction-diffusion system, Computers ＆ Mathematics with Applications, Vol. 81, pp. 93-97.
G.Adomian, 1998, “Solutions of nonlinear P.D.E., Applied Mathematics Letters, Vol. 11, No. 3, pp. 121-123.
G.Adomian, 1998, “Nonlinear dissipative wave equation, Applied Mathematics Letters, Vol. 11, No. 3, pp. 125-126.
G. Domairry, M. Fazeli, 2009 ,“Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity,Communications Nonlinear SciSimulation, vol.14,pp.489-499
J. Fadaei , 2011,“Application of Laplace-Adomian Decomposition Method on Linear and Nonlinear System of PDEs ,“Applied Mathematical Sciences , Vol. 5,no.27,pp.1307-1315
J.H.Kuang, C.J.Chen, 2005, “Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators, Mathematical and Computer Modelling, Vol. 41, pp. 1479-1491.
M.Torabi , H.Yaghoobi, 2012 ,Analytical Approaches for Thermal Analysis of Radiative Fin with a Step Change in Thickness and Variable Thermal Conductivity , Heat Transfer-Asian Research , vol .41(4)
M.Torabi, A.Aziz, K.Zhang, 2012 , “A comparative study of longitudinal fins of rectangular, trapezoidal and concave parabolic profiles with multiple nonlinearities , Energy,vol.51, pp.243-256
M.Torabi, H.Yaghoobi, A.Aziz, 2012 ,Analytical Solution for Convective-Radiative Continuously Moving Fin with Temperature-Dependent Thermal Conductivity , vol.33 ,pp.924-941
M. Hatami ,D.D. Ganji, 2013 ,Thermal performance of circular convective–radiative porous fins with different section shapes and materials, Energy Conversion and Management, vol.76,pp.185-193
M.Y. Ongun , 2010 , “The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+T cell, Mathematical and Computer Modelling , vol. 53 ,pp. 597-603
P.Y. Tsai , 2010 , “Application od the Hybrid Laplace Adomian
Decomposition Method to Nonlinear Physcial Systems,
S.A.Khuri , 2001, “On the decomposition method for approximate solution of nonlinear ordinary differential equation, International Journal of Mathematical Education in Science and Technology, Vol. 32, No. 4, pp. 525-539.
S.A.Khuri, 2004, “A new approach to Bratu’s equation Applied Mathematics and Computation, Vol. 147, pp. 131-136.
S.A. Khuri , 2011 , A Laplace decomposition algorithm applied to a class of nonlinear differential equations Journal of Applied Mathematics 1:4,pp.141-155

S. Saedodin , S. Mohammad, B. Motaghedi, 2014,Comprehensive analytical study for convective-radiative continuously moving plates with multiple non-linearities Energy Conversion and Management,Vol.88,pp.160-168
S.B. Coşkun, M.T. Atay , 2008 ,Fin efficiency analysis of convective straight fins with temperature dependent thermal conductivity using variational iteration method Applied Thermal Engingeering ,vol.28,pp.2345-2352
S.N. Venkatarangan, and K. Rajalakshmi,1995, “A modification of Adomian’s solution for nonlinear oscillatory systems, Computers ＆ Mathematics with Applications, Vol. 29, No. 6, pp. 67-73.
Y.T. Yang ,C.C. Chang , C.K. Chen, 2011 , “A double decomposition method for solving the annular hyperbolic profile fins with variable thermal conductivity Heat Transfer Engineering, Vol. 31,Issue 14, pp. 1165-1172.