臺灣博碩士論文加值系統

本文旨在利用Laplace Adomian 分解法(Laplace Adomian decomposition method，簡稱LADM) 結合Pad?近似法(Pad? approximant)來求解非線性系統中的近似解析解。 LADM即結合Laplace轉換 (Laplace transformation) 與George Adomian教授所提出的Adomian分解法(Adomian decomposition method，簡稱ADM) 所衍生出來的一個方法。但是無論ADM或LADM所求得的近似解析解為一個無窮級數解的形式，除非求到無窮多項，否則所得到的解只是一個截斷級數解(truncated series solution)，而且會很快的發散而無法收斂。Pad?近似法可以解決此問題以獲得更佳的精確度與收斂性。
Laplace Adomian混合分解法對於求解非線性系統時，不需要線性化(linearization)或微小參數(small parameter)之假設之限制，便可以找出正確的數學模型來呈現一個真實的非線性物理系統。利用Laplace Adomian混合分解法求解非線性物理系統時，不但可以延伸其定義域的範圍，以解決截斷級數解無法收斂的問題，並且在求解邊界條件問題時，亦不需要轉換成初始值問題即可直接求解。此外，LADM-Pad?近似混合法可以以代數的形式來表示一個非線性系統的近似解析解，不需要向數值解之方法，在每一次運算時皆需要把所有的值代入才可以計算。
本文中提出一個Laplace Adomian混合分解法，即結合LADM與Pad?近似法的混合法(LADM-Pad? approximant technique)，針對一些非線性問題做探討，對於非線性單擺、振動、控制與流體系統所得到的計算結果，與文獻中的解析解與不同方法的數值解作比較，結果相當吻合，足以證明LADM-Pad?近似混合法相對於Adomian分解法來說，是一個簡單、精確、有效且快速收斂的方法。

The Laplace Adomian decomposition method (LADM) combines the numerical Laplace transform algorithm and the Admoian decomposition method (ADM). The truncated series solution solved by the LADM diverges rapidly as the applicable domain increases. However, the Pad? approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. In this paper, a hybrid method of the LADM combined with the Pad? approximant, named the hybrid Laplace Adomian decomposition method is proposed to solve the nonlinear physical systems to demonstrate efficient and reliable results.
The linearization and small parameter assumptions are unnecessary for solving the nonlinear system problems by the hybrid Laplace Adomian decomposition method. The LADM─Pad? approximant solution is easy to obtain to demonstrate a real nonlinear physical phenomenon, and the transformation of the boundary value conditions into an initial value problem is also unnecessary when solving a boundary value problem. Furthermore, the LADM─Pad? approximant solution is able to demonstrate a nonlinear physical system by an algebra form. So the calculation is no like the numerical method that every value needs to be known every time.
The hybrid Laplace Adomian decomposition method has been successfully applied to solve various nonlinear problems such as, nonlinear pendulum systems, nonlinear oscillation systems, nonlinear control systems, and nonlinear fluid dynamic systems. The LADM-Pad? approximant solutions demonstrate efficient and reliable results and have been shown a good accuracy and convergence in comparison with the exact solutions and other numerical method solutions. Moreover, the LADM─Pad? approximant solutions have been demonstrated not only the superiority of the accuracy and convergence over both the ADM and LADM solutions, but also extended the applicable domain to overcome their drawbacks.

中文摘要 I
英文摘要 III
誌謝 V
目錄 VII
表目錄 XI
圖目錄 XII
符號說明 XV

第一章 緒論 1
1-1前言 1
1-2文獻回顧 3
1-2-1 非線性物理系統之發展 3
1-2-2 Adomian分解法 5
1-3本文架構 9
第二章 Laplace Adomian混合分解法 11
2-1 Adomian分解法 12
2-2 Adomian多項式 16
2-2-1 非線性(nonliinear)多項式 16
2-2-2非線性微分(derivative)多項式 18
2-2-3三角(trigonometric)函數多項式 20
2-2-4雙曲線(hyperbolic)函數多項式 23
2-2-5指數(exponential)函數多項式 27
2-2-6對數(logarithmic)函數多項式 28
2-3 修正Adomian分解法 29
2-3-1修正ADM（一） 30
2-3-2修正ADM（二） 34
2-4 Laplace Adomian分解法 37
2-4-1運算法則 37
2-4-2討論 39
2-5 Pad? 近似法 44
2-6 LADM-Pad?近似混合法 46
2-6-1 LADM-Pad?近似混合法（一） 47
2-6-2 LADM-Pad?近似混合法（二） 50
第三章 非線性振動系統之分析 53
3-1單擺(pendulum) 55
3-1-1 小振幅簡單單擺 57
3-1-2 大振幅簡單單擺 62
3-1-3 具阻尼單擺系統 66
3-2 Duffing振盪系統 72
3-3 Van der Pol振盪系統 77
3-4 Rayleigh振盪系統 82
3-5 Helmholtz振盪系統 85
3-6結論 89
第四章 非線性控制系統之分析 91
4-1 Riccati微分方程（一） 93
4-2 Riccati微分方程（二） 97
4-3 Riccati微分方程（三） 102
4-4 結論 105
第五章 非線性流體系統之分析 107
5-1 邊界層流問題之數學模型建立 108
5-2 邊界層流之流場問題 111
5-3 邊界層流之熱傳問題 117
5-4 結論 125
第六章 結論與建議 127
6-1 綜合結論 127
6-2 未來研究方向之建議 130
參考文獻 131
自述 140

Abbasbandy, S., 2006, “Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Applied Mathematics and Computation, Vol. 172, pp. 485-490.
Abbasbandy, S., 2006, “Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Applied Mathematics and Computation, Vol. 175, pp. 581-589.
Abbasbandy, S., 2007, “A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method, Chaos, Solitions and Fractals, Vol. 31, pp. 257-260.
Abbasbandy, S., 2007, “A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Journal of Computational and Applied Mathematics, Vol. 207, pp. 59-63.
Adomian, G., 1994, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Academic Publishers, Boston.
Adomian, G., 1995, “Delayed nonlinear dynamical systems, Mathematical and Computer Modelling, Vol. 22, No. 3, pp. 77-79.
Adomian, G., 1995, “Stochastic Burgers’ equation, Mathematical and Computer Modelling, Vol. 22, No. 8, pp. 103-105.
Adomian, G., 1995, “Fisher-Kolmogorov equation, Applied Mathematics Letters, Vol. 8, No. 2, pp. 51-52.
Adomian, G., 1996, “Nonlinear Klein-Gordon equation, Applied Mathematics Letters, Vol. 9, No. 3, pp. 9-10.
Adomian, G., 1996, “Nonlinear random vibration, Applied Mathematics and Computation, Vol. 77, pp. 109-112.
Adomian, G., 1996, “Solution of the coupled nonlinear partial differential equations by decomposition, Computers ＆ Mathematics with Applications, Vol. 31, No. 6, pp. 117-120.
Adomian, G., 1996, “The fifth-order Korteweg-de Vries equation, International Journal of Mathematics and Mathematical Sciences, Vol. 19, No. 2, p. 415.
Adomian, G., 1997, “On the dynamics of a reaction-diffusion system, Computers ＆ Mathematics with Applications, Vol. 81, pp. 93-97.
Adomian, G., 1998, “Solutions of nonlinear P.D.E., Applied Mathematics Letters, Vol. 11, No. 3, pp. 121-123.
Adomian, G., 1998, “Nonlinear dissipative wave equation, Applied Mathematics Letters, Vol. 11, No. 3, pp. 125-126.
Allan, F.M. and Syam, M.I., 2005, “On the analytic solutions of the nonhomogeneous Blasius problem, Journal of Computational and Applied Mathematics, Vol. 22, No. 182, pp. 362-371.
Adomian, G., 1998, “Solution of the Thomas-Fermi equation, Applied Mathematics Letters, Vol. 11, No. 3, pp. 131-133.
Bacon, M.E. and Nguyen, D.D., 2005, “Real-world damping of a physical pendulum, European Journal of Physics, Vol. 26, pp. 651-655.
Bejan, A., 2004, Convection heat transfer, John Wiley ＆ Sons, New Jersey.
Baker, Jr., G.A. and Graves-Morris, P., 1996, PAD? APPROXIMANTS, Cambridge University Press, 2nd edition, New York.
Baker, G.L. and Blackburn, J.A., 2005, The Pendulum, Oxford University Press, New York.
Bel?ndez, A., Hern?ndez, A., M?rquez, A., Bel?ndez, T., and Neipp, C., 2006, “Analytical approximations for the period of a nonlinear pendulum, European Journal of Physics, Vol. 27, pp. 539-551.
Bel?ndez, A., Hern?ndez, A., Bel?ndez, T., Neipp, C., and M?rquez, A., 2007, “Application of the homotopy perturbation method to the nonlinear pendulum, European Journal of Physics, Vol. 28, pp. 93-104.
Bulut, H. and Evans, D.J., 2002, “On the solution of the Riccati equation by the decomposition method, International Journal of Computer Mathematics, Vol. 79, No.1, pp. 103-109.
Butikov, E.I., 1999, “The rigid pendulum - an antique but evergreen physical model, European Journal of Physics, Vol. 20, pp. 429-441.
Cadwell, L.H. and Boyko, E.R., 1991, “Linearization of the simple pendulum, American Journal of Physics, Vol. 59, No. 11, pp. 979-981.
Chen, W. and Lu, Z., 2004, “An algorithm for Adomian decomposition method, Applied Mathematics and Computation, Vol. 159, pp. 221-235.
Chiu, C.H. and Chen, C.K., 2002, “A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International of Heat and Mass Transfer, Vol. 45, pp. 2067-2075.
Cortell, R., 2005, “Numerical solutions of the classical Blasius flat-plate problem, Applied Mathematics and Computation, Vol. 170, pp. 706-710.
Dehghan, M., Shakourifar, M., and Hamidi, A., 2009, “The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique, Chaos, Solitions and Fractals, Vol. 39, pp. 2509-2521.
El-sayed, S.M. and Abdel-Aziz, M.R., 2003, “A comparison of Adomian’s decomposition method and wavelet-Galerkin method for solving integro-differential equations, Applied Mathematics and Computation, Vol. 136, pp. 151-159.
El-Tawil, M.A., Bahnasawi, A.A., and Abdel-Naby, A., 2004, “Solving Riccati differential equation using Adomian’s decomposition method, Applied Mathematics and Computation, Vol. 157, pp. 503-514.
Fulcher, L.P. and Davis, B.F., 1976, “Theoretical and experimental study of the motion of the simple pendulum, American Journal of Physics, Vol. 44, No. 1, pp. 51-55.
Ganley, W.P., 1985, “Simple pendulum approximation, American Journal of Physics, Vol. 53, No. 1, pp. 73-76.
G?lsu, M. and Sezer, M., 2006, “On the solution of the Riccati equation by the Taylor matrix method, Applied Mathematics and Computation, Vol. 176, pp. 414-421.
Gough, W., 1983, “The period of a simple pendulum is not 2π(L/g)1/2, European Journal of Physics, Vol. 4, pp. 53-54.
Abdel-Halim Hassan, I.H., 2008, “Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos Fractals ＆ Solitions, Vol. 36, pp. 53-65.
He, J.H., 1998, “Approximate analytical solution of Blasius’ equation, Communications in Nonlinear Science ＆ Numerical Simulation, Vol. 3, pp. 260-363.
He, J.H., 2003, “A simple perturbation approach to Blasius equation, Applied Mathematics and Computation, Vol. 140, pp. 217-222.
Hsu, J.C., Lai, H.Y., and Chen, C.K., 2008, “Free vibration of non-uniform Euler-Bernoulli beams with general elastically end constraints using Adomian modified decomposition method, Journal of Sound and Vibration, Vol. 318, pp. 965-981.
Howarth, L., 1938, “On the solution of the laminar boundary layer equations, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 164, pp. 547-579.
Jiao, Y.C., Yamamoto, Y., Dang, C., and Hao, Y., 2002, “An aftertreatment technique for improving the accuracy of Adomian’s decomposition method, Computers ＆ Mathematics with Applications, Vol. 43, pp. 783-798.
Khuri, S.A., 2001, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, Vol. 1, No. 4, pp. 141-155.
Khuri, S.A., 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.
Khuri, S.A., 2004, “A new approach to Bratu’s equation, Applied Mathematics and Computation, Vol. 147, pp. 131-136.
Kuang, J.H. and Chen, C.J., 2005, “Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators, Mathematical and Computer Modelling, Vol. 41, pp. 1479-1491.
Kuo, B.L., 2003, “Application of the differential transformation method to the solutions of Falkner-Skan wedge flow, Acta Mechanica, Vol. 165, pp.161-174.
Kuo, B.L., 2004, “Thermal boundary-layer problems in a semi-infinite flat plate by the differential transformation method, Applied Mathematics and Computation, Vol. 150, pp.303-320.
Lewowski, T. and Wo?niak, K., 2002, “The period of a pendulum at large amplitudes: a laboratory experiment, European Journal of Physics, Vol. 23, pp. 461-464.
Liao, S.J., 1999, “An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, International Journal of Non-Linear Mechanics, Vol. 34, pp. 759-778.
Momani, S., 2004, “Analytical approximate solutions of nonlinear oscillators by the modified decomposition method, International Journal of Modern Physics C, Vol. 15, No. 7, pp. 967-979.
Momani, S. and Shawagfeh, N., 2006, “Decomposition method for solving fractional Riccati differential equations, Applied Mathematics and Computation, Vol. 182, pp. 1083-1092.
Momani, S. and Ert?rk, V.S., 2008, “Solutions of non-linear oscillators by the modified differential transform method, Computers and Mathematics with Applications, Vol. 55, pp. 833-842.
?zis, T. and Y?ld?r?m, A., 2008, “Comparison between Adomian’s decomposition method and He’s homotopy perturbation method, Computers ＆ Mathematics with Applications, Vol. 56, pp. 1216-1224.
Parwani, R.R., 2004, “An approximate expression for the large angle period of a simple pendulum, European Journal of Physics, Vol. 25, pp. 37-39.
Ramos, J.I., 2008, “Series approach to the Lane-Emden equations and comparison with the homotopy perturbation method, Chaos Fractals ＆ Solitions, Vol. 38, pp. 400-408.
Ranasinghe, A.I., 2009, “Solution of Blasius equation by decomposition, Applied Mathematical Sciences, Vol. 3, pp. 605-611.
Squire, P.T., 1986, “Pendulum damping, American Journal of Physics, Vol. 54, No. 11, pp. 984-991.
Syam, M.I. and Hamdan, A., 2006, “An efficient method for solving Bratu equations, Applied Mathematics and Computation, Vol. 176, pp. 704-713.
Tan. Y. and Abbasbandy, S., 2008, “Homotopy analysis method for quadratic Riccati differential equation, Communications in Nonlinear Science and Numerical Simulation, Vol. 13, pp. 539-546.
Tang, B.Q. and Li, X.F., 2007, “A new method for determining the solution of Riccati differential equations, Applied Mathematics and Computation, Vol. 194, pp. 431-440.
Tien, W.C. and Chen, C.K., 2009, “Adomian decomposition method by Legendre polynomials, Chaos, Solitons and Fractals, Vol. 39, pp. 2093-2101.
Van der Pol, B. and Van der Mark, J., 1927, “Frequency demultiplication, Nature, Vol. 120, No. 3019, pp. 363-364.
Venkatarangan, S.N. and Rajalakshmi, K., 1995, “A modification of Adomian’s solution for nonlinear oscillatory systems, Computers ＆ Mathematics with Applications, Vol. 29, No. 6, pp. 67-73.
Wang, L., 2004, “A new algorithm for solving classical Blasius equation, Applied Mathematics and Computation, Vol. 157, pp.1-9.
Wazwaz, A.M., 1998, “A comparison between Adomian decomposition method and Taylor series method in the series solutions, Applied Mathematics and Computation, Vol. 97, pp.37-44.
Wazwaz, A.M., 1999, “Analytical approximations and Pad? approximants for Volterra’s population model, Applied Mathematics and Computation, Vol. 100, pp.13-25.
Wazwaz, A.M., 1999, “A reliable modification of Adomian decomposition method, Applied Mathematics and Computation, Vol. 102, pp. 77-86.
Wazwaz, A.M., 1999, “The modified decomposition method and Pad? approximants for solving the Thomas–Fermi equation, Applied Mathematics and Computation, Vol. 105, pp.11-19.
Wazwaz, A.M., 2000, “A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation, Vol. 111, pp.53-69.
Wazwaz, A.M., 2001, “A reliable algorithm for solving boundary layer problems for higher-order integro-differential equations, Applied Mathematics and Computation, Vol. 118, pp. 327-342.
Wazwaz, A.M. and El-sayed, S.M., 2001, “A new modification of the Adomian decomposition method for linear and nonlinear operators, Applied Mathematics and Computation, Vol. 122, pp. 393-405.
Wazwaz, A.M., 2001, “A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Computers ＆ Mathematics with Applications, Vol. 41, pp.1237-1244.
Wazwaz, A.M., 2002, “A new method for solving singular initial value problems in second-order ordinary differential equations, Applied Mathematics and Computation, Vol. 128, pp. 45-57.
Wazwaz, A.M. and Gorguis, A., 2004, “An analytic study of Fisher’s equation by using Adomian decomposition method, Applied Mathematics and Computation, Vol. 154, pp. 609-620.
Wazwaz, A.M., 2005, “Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Applied Mathematics and Computation, Vol. 161, pp. 543-560.
Wazwaz, A.M., 2005, “Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation, Vol. 166, pp. 652-663.
Wazwaz, A.M., 2006, “Pad? approximants and Adomian decomposition method for solving the Flierl–Petviashivili equation and its variants, Applied Mathematics and Computation, Vol. 182, pp. 1812-1818.
Wazwaz, A.M., 2007, “The variational iteration method for solving two forms of Blasius equation on a half-infinite domain, Applied Mathematics and Computation, Vol. 188, pp. 485-491.
Yu, L.T. and Chen, C.K., 1998, “The solution of the Blasius equation by the differential transformation method, Mathematical and Computer Modelling, Vol. 28, pp. 101-111.