運算矩陣(operational matrix)的觀念被廣泛地應用在許多領域，像是解微分方程、系統參數鑑別，以及求解線性系統的最佳化等。運算矩陣的主要特性是其可以將微分方程式轉換為代數(algebra)的形式，如此不但可以簡化問題而且可以加快計算的速度。不過傳統上運算矩陣的推導方法不但費時而且並不統一(unified)。本論文提出一個新的統一運算矩陣推導方法，來推導正交函數的積分或微分運算矩陣。該方法有許多優點，包括簡單、效果好及電腦計算導向，而且最重要的是統一的特性，這使得我們能使用許多的正交函數。

本論文以運算矩陣為基礎，發展了一個新的數值方法求解逆拉氏轉換(Laplace transform)的問題。這個新方法只需要矩陣的乘法及一般的代數運算。這表示線性非時變系統能完全以代數方法來計算其數值解。所提出的方法比傳統的查表(dictionary-type)法或輪廓積分(contour integration)方法都還要簡單。

正交函數運算矩陣有一個有用的特性，就是高次方的積分運算矩陣很快趨近於零矩陣。本論文以正交函數運算矩陣為基礎發展了一個新的模式縮減(model reduction)方法。它刪除較高次方的積分運算矩陣項，而得到簡化或縮減的模式。從實驗結果中得知我們的方法成效良好。

在電腦分析，或是模擬實體系統像是流體力學、電磁學及人類生理學等領域中，偏微分方程是很主要的方法。本論文提出一個新的運算矩陣方法來求解偏微分方程。它能夠將問題簡化成一個里亞普諾夫矩陣方程式(Lyapunov matrix equation)。我們所提出的方法優點為：計算簡單且是電腦計算導向、應用範圍廣泛及不會出現數值解不穏定的問題。

分數微積分(fractional calculus)包括了非整數(fractional)次方的微分與積分。而含有分數運算子，像是 和 ，的微分方程稱為分數微分方程(fractional differential equation)。它們在科學及工程上有很多的應用，但是通常它們只在某些限制條件下才有解析解，而且也很難以數值方法來求解。本論文中提出了一個以正交函數運算矩陣為基礎的數值方法來解這一類的問題。實驗結果顯示這個新方法較傳統方法簡單而且電腦計算導向。

The idea of the operational matrix has been widely applied to many fields such as solving the differential equations, identifying and optimizing linear systems etc. The main character of the operational method is to convert a differential equation into an algebraic one, it not only simplifies the problem but also speed up the computation. But the conventional method of deriving the operational matrix is not only time-consuming but also non-unified. In this thesis, we propose a new unified method to deriving the operational matrix of orthogonal functions for integration and differentiation. It can derive the operational matrix with not only the integer order but also the fractional order (non-integer order). The proposed method has several advantages, it is simple, efficient, computer-oriented and the most important feature is that it is unified, in which many orthogonal functions can be used.

Based on the derived operational matrix, we present a new method for performing numerical inversion of the Laplace transform. Only matrix multiplications and ordinary algebraic operations are involved in the new approach. This implies that linear time-invariant systems can be completely solved algebraically and numerically. The proposed method is much simpler as compared with the dictionary-type method and the contour integration method. By applying the proposed method to various types of functions, hundreds of the Laplace transform pairs can be established for practical use.

The operational matrix of orthogonal functions have a useful property, it approaches the zero matrix very fast as the order is increased. In this thesis, we introduce a new model reduction method based on the operational matrix of orthogonal functions. Following Nixon’s philosophy, we simply drop the higher order terms in order to obtain various simplified or reduced models. From the experiment, it shows that our approach obtains good results.

Partial differential equations are at the heart of many, if not most, computer analyses or simulations of continuous physical systems, such as fluids, electromagnetic fields, the human body, and so on. In this thesis, we propose a new operational method to solve the partial differential equations numerically. It transforms the problem to a simple Lyapunov matrix equation solving. Two partial differential equations solved by the new method are demonstrated. Advantages of the proposed method include (1) the computation is simple and computer oriented, (2) the scope of application is wide and (3) the step size used could be large and the result obtained is still satisfactory. The numerically unstable problem never occurs in our method.

Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as and is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also the numerical methods are difficult to solve. In this thesis, we propose a new numerical method based on the operational matrix of orthogonal functions to solve this type of problems. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods.

[1] C. F. Chen and C.H. Hsiao, “Design of piecewise constant gains for optimal control via Walsh function,” IEEE Transactions on Automatic Control, AC-20 (5), pp.596-603, 1975.
[2] C. F. Chen, Y. T. Tsay, and T. T. Wu, “Walsh operational matrices for fractional calculus and their application to distributed system,” J. Franklin Inst., vol. 303, pp. 267-284, Mar. 1977.
[3] W. L. Chen and Y. P. Shih, “Parameter estimation of bilinear systems via Walsh functions,” J. Franklin Inst., vol. 305, pp.249-257, May 1978.
[4] P. Sannuti, “Analysis and synthesis of dynamic system via block-pulse functions,” Proc. IEE, vol. 124, pp.569-571, June 1977.
[5] B. Cheng and N. S. Hsu, “Analysis and parameter estimation of bilinear systems via block-pulse functions,” Int. J. Contr., vol.36, pp.53-65, 1982.
[6] C. Hwang and Y. P. Shih, “Parameter identification via Laguerre polynomials,” Int. J. Syst. Sci., vol.13, pp.209-217, 1982.
[7] R. Y. Chang and M. L. Wang, “Parameter identification via shifted Legendre polynomials,” Int. J. Syst. Sci,. vol. 13, pp.1125-1135, 1982.
[8] Y. M. Shih, “Application of Chebyshev Polynomials in analysis and identification of linear systems,” J. Chin. Inst. Engrs., vol. 6, pp.135-140, Apr. 1983.
[9] S. G. Mouroutsos and P. D. Sparis, “Taylor series approach to system identification, analysis and optimal control”, J. Franklin Inst., vol. 319, pp.359-371, Mar. 1985.
[10] P. N. Paraskevopoulos, P. D. Sparis, and S. G. Mouroutsos: “The Fourier series operational matrix of integration”, Int. J. System Sci., 16 (2), pp. 171-176, 1985.
[11] C. F., Chen and C. H. Hsiao: “Haar wavelet method for solving lumped and distributed-parameter systems”, IEE Proc. Control Theory and Application., 144 (1), pp. 87-94, January 1997.
[12] F. Tricomi: “Transformazione di Laplace e polinami di Laguerre”, R. C. Accad. Nat. dei Lincei 21, pp.232-239, 1935.
[13] D. V. Widder: “An Application of Laguerre polynomials”, Duke Math. J. 1, pp.126-136, 1935.
[14] G. Doetsch: Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York, 1974.
[15] B. Van Der Pol and H. Bremmer: Operational Calculus, Cambridge Univ. Press, 1955. (Reprinted in 1987 by Chelsea Press, New York).
[16] E. Hille: Analytic Function Theory, Vol. 2, Ginn, Boston, 1962.
[17] W. Kaplan: Operational Methods, Addison, Reading, 1962.
[18] G. Moretti: Functions of a Complex Variable, Prentice Hall, Englewood Clis, 1964.
[19] E. E. Ward: “The calculation of transients in dynamical systems”, Proc. Camb. Phil. Soc. 50, pp.49-59, 1954.
[20] C. F. Chen: “A new formula for obtaining the inverse Laplace transformation in terms of Laguerre functions”, IEEE Intern. Convention Record 14, pp.281-287, 1966.
[21] R. A. Spinelli: “Numerical inversion of a Laplace transform”, SIAM J. Numer. Anal. 3, pp.636-649, 1966..
[22] W. T. Weeks: “Numerical inversion of Laplace transforms using Laguerre functions”, J. ACM 13, pp.419-426, 1966.
[23] Y. L. Luke: The Special Functions and Their Approximations, vol. 2, Academic, New York, 1969.
[24] R. Piessens and M. Branders: “Numerical inversion of the Laplace transform using generalized Laguerre polynomials”, Proc. IEEE 118, pp.1517-1522, 1971.
[25] B. Davies and B. L. Martin: “Numerical inversion of the Laplace transform: A survey and comparison of methods”, J. Comp. Phys. 33, pp.1-32, 1979.
[26] H. Weber: “Numerical computation of the Fourier transform using Laguerre functions and the fast Fourier transform”, Numer. Math. 36, pp.197-209, 1981.
[27] W. T. Wu and H. R. Ong: “On a functional approximation for inversion of Laplace transforms”, J. Chinese Inst. Eng. 6, pp.265-268 , 1983.
[28] J. N. Lyness and G. Giunta: “A modication of the Weeks method for numerical inversion of the Laplace transform”, Math. Comp. 47, pp.313-322, 1986.
[29] B. S. Garbow, G. Giunta, J. N. Lyness and A. Murli: “Algorithm 662: A FORTRAN software package for numerical inversion of the Laplace transform based on Weeks' method”, ACM Trans. Math. Software 14, pp.171-176, 1988.
[30] B. S. Garbow, G. Giunta, J. N. Lyness and A. Murli: “Software for an implementation of Weeks' method for the inverse Laplace transform problem”, ACM Trans. Math. Software 14, pp.163-170, 1988.
[31] D. G. Duffy: “On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications”, ACM Trans. Math. Software 19, pp.333-359, 1993.
[32] C. F. Chen and C. H. Hsiao: “A Haar wavelet operational matrix for solving dynamic systems”, Electromagnetic Research Symposium Proceedings, Seattle, WA., July 1995.
[33] W. H. Press, S. A. Teukolsky, W. T. Vettering and B. P. Flannery: Numerical Recipes in C, the art of Scientific computing, second edition, Cambridge University Press, 1988.
[34] K. B. Oldham and J. Spanier: The Fractional Calculus. New York, Academic, 1974.
[35] S. G. Samko, A. A. Kilbas, and O. I. Marichev: Fractional Integrals and Derivatives, Theory and Applications. Langhorne, PA: Gordon Breach Sci. Publ., 1993.
[36] A. C. McBride and G. F. Roach: Fractional Calculus, Research Notes in Mathematics, Boston, MA: Pitman Adv. Publ., 1985.
[37] V. Kiryakova: “Generalized Fractional Calculus and Its Applications,” Pitman Res. Notes Mathem. Ser. 301, Longman Scientific Tech., Long-man Group, Essex, UK, 1994.
[38] H. T. Davis: The Theory of Linear Operators. Bloomington, IN: Principia Press, pp. 276-292, 1936.
[39] E. R. Love: “Fractional Derivatives of imaginary order”, J. London Math. Soc., Part 2, 2nd ser., vol. 3 pp.241-259, Feb. 1971.
[40] N. Engbeta, “On Fractional Calculus and Fractional Multipoles in Electromagnetism”, IEEE transactions on antennas and propagation, vol.44, no. 4, pp.554-565, April 1996.
[41] J. Luzen: “Liouville’s differential calculus of arbitrary order and its electrodynamical origin”, Proc. 19th Nordic Congress Mathematicians, Reykjavik, Icelandic Mathematical Soc., pp.149-160, 1985.
[42] R. E. Bellman, and R. E. Kalaha: Modern Analytic and Computational Methods in Science and Mathematics, American Elsevier Publishing Co. Inc., New York, 1966.
[43] P. J. Nakin: Oliver Heaviside, IEEE Press, New York, pp. 227-230, 1988.
[44] A. Haar: “Zur throrieder orthogonal function system”, Math. Ann., 69, pp. 331-371, 1910.
[45] A. N. Akansu and, R. A. Haddad: Multi-resolution Signal Decomposition: transform, sub-bands and wavelets, Academic Press, Inc., pp.60-61, 1981.
[46] M. Vetterli and J. Kovacevic: Wavelets and Sub-band Coding, Prentice-Hall PTR, Englewood Cliffs, New Jersey, pp.32-34, 1995.
[47] A. V. Oppenheim and R. W. Schafer: Discrete-time Signal Processing, Prentice Hall International Inc., London, 1989.
[48] R. N. Bracewell, The Hartley Transform, Oxford University Press, New York, 1986.
[49] S. L. Manney, M. S. Nakhla: “Analysis of Nonuniform, Frequency-Dependent High-Speed Interconnects Using Numerical Inversion of Laplace Transform,” IEEE Transactions on Computer-Aided Design of Integrated circuits and Systems, vol. 13, no. 12 December, pp.1213-1525, 1994.
[50] E. C. Chang and S. M. Kang: “Computationally Efficient Simulation of a Lossy Transmission Line with Skin Effect by Using Numerical Inversion of Laplace Transform”, IEEE Transactions on circuits and systems I: Fundamental Theory and applications, vol. 39, no. 11, Nov., pp.861-868, 1992.
[51] T. V. Nguyen, “Efficient Simulation of Lossy and Dispersive Transmission Lines”, IBM Microelectronics, Hopewell Junction NY.
[52] V. Biolkova and D. Biolek, “Time domain analysis of linear systems using state space approach”
[53] D. G. Duffy, “On the Numerical Inversion of Laplace Transforms: Comparison of Three New Methods on Characteristic Problems from Applications”, ACM Transactions on Mathematical Software, Vol.19, No.3, pp.333-359, September 1993.
[54] B. Davies and B. Martin: “Numerical inversion of the Laplace transform: A survey and comparison of methods”, J. Comput. Phys. 33, 1, pp.1-32, Oct. 1979.
[55] N. W. Mclachlan: “Complex Variables and Operational Calculus”, 2nd ed., Macmillan, New York, 1953.
[56] T. Huddleston, “Numerical Inversion of Laplace Transforms,” April 1999.
[57] G. Hong and U. Hirdes, “Algorithm 27. A method for the numerical inversion of Laplace transforms”, J. Comput. Appl. Math. 10, pp.113-132, Jan. 1984.
[58] K. S. Crump, “Numerical Inversion of Laplace Tramsforms Using a Fourier Series Approximation”, J. Assoc. Comput. Machinery, 23, pp.89-96, 1976.
[59] A. Talbot, “The accurate numerical inversion of Laplace transforms,” J. Inst. Math. Appl. 23, 1, pp.97-120, Jan 1979.
[60] A. Murli and M. Rizzardi, “Algorthm 682. Talbot’s method for the Laplace inversion problem”, ACM Trans. Math. Softw. 16, 2, pp.158-168, June 1990.
[61] R. G. Richard, and D Duong: Applied Mathematics And Modeling for Chemical engineers, John Wiley & Sons, Inc, 1995.
[62] V. Zakian: “Numerical inversion of Laplace Transform”, Electronics Letters, 5 No. 6, pp 120-121 March 20, 1969.
[63] V. Zakian: “Least-Squares Optimization of Numerical Inversion of Laplace Transforms”, Electronics Letters Vol. 7, pp 71-72, 1971.
[64] V. Zakian: “Optimization of Numerical Inversion of Laplace Transforms”, Electronics Letters, Vol 6, No. 21, pp 677-679, October, 1970.
[65] W. T. Weeks, “Numerical inversion of Laplace transforms using Laguerre functions”, J. ACM 13, 3, pp.419-429, July 1966.
[66] J. N. Lyness and G. Giunta, “A modification of the Weeks method for numerical inversion of the Laplace transform,” Math. Comput. 47, 175, pp.313-322, July 1986.
[67] B. S. Garbow, G. Gjunta, J. N. Lyness and A. Murli, “Software for an implementation of Week’s method for the inverse Laplace transform problem”, ACM trans. Math. Softw. 14, 2, pp.163-170, June 1988.
[68] B. S. Garbow, G. Gjunta, J. N. Lyness and A. Murli, “Algorithm 662: A Fortran software package for the numerical inversion of the Laplace transform based on Weeks’ method,” ACM Trans. Math. Sofw. 14, 2, pp.171-176, June 1988.
[69] M. F. Gardner, and J. L.: Barnes: “Transient in Linear Systems” Vol.1, John Wiley & Sons, Inc., New York, 1942.
[70] R. H. Bartels and G. W. Stewart: ‘Solution of the Matrix Equation AX+XB=C’, Comm. of The ACM, 15 (9), 1972.
[71] M. D. Ortigueira, Introduction to fractional linear systems. Part 1: Continuous-time case.
[72] R. N. Kalia: Recent advances in fractional calculus, Global Publishing Company, 1993.
[73] K. S. Miller and B. Ross, “An introduction to the fractional calculus and fractional differential equations”, Wiley, 1993.
[74] S. G. Samko, A. A. Kukbas and O. I. Marichev: “Factional integrals and derivatives---theory and applications”, Gordon and Breach Science Publishers, 1987.
[75] L. M. C. Campos, “On a concern of derivative of complex order with applications to special functions”, IMA. J. Appl. Math, 33, pp.109-133, 1984.
[76] L. M. C. Campos, “ On the solution of some simple fractional differential equations”, Int. J. Math. Sci. 13(3), pp.481-496, 1990.
[77] K. Nishimoto, Fractional calculus, Descartes Press Co., Koriyama, Japan, 1989.
[78] C. G. Koh and J. M. Kelly, “Application of fractional derivatives to seismic analysis of based-isolated models,” Earthq. Eng. Siruct. Dyn., 19, pp.229-241, 1990.
[79] B. B. Mandelbrot, “The fractal geometry of nature,” W. H. Freeman and Company, New York, 1983
[80] B. B. Mandelbrot and J. W. Vanness, “The fractional Brownian motions, fractional noises and applications”, SIAM Rev., 10, pp.4, 1968.
[81] I. Petras, B. M. Vinagre, L. Dorcak and V. Feliu, “Fractional Digital Control of a Heat Solid: Experimental Results”, International Carpathian Control Conference ICCC’ 2002 Malenovice, Czech Republic. pp.365-370, May 2002.
[82] S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional Integrals and Derivatives: theory and applications” Gordon and Breach Science Publishers, 1993.
[83] K. S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Son, New York, 1993.
[84] K. B. Oldham and J. Spanier: The Fractional Calculus, Academic Press, New York, pp.1-15, 1974.
[85] F. Mainardi: Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. CISM Lecture Notes, Udine, Italy, 1996.
[86] M. Axtell and E. M. Bise: “Fractional Calculus Applications in Control Systems”,.Proc. of the IEEE Nat. Aerospace and Electronics Conf., New York, pp. 563-566, 1990.
[87] Doral’L: “Numerical Models for Simulation the Fractional-Order Control”, Systems. UEF SAV, The Academy of Sciences, Inst. of Experimental Physics, Košice, Slovak Republic. 1994.
[88] A. Oustaloup, La Dérivation non Entiere. HERMES, Paris. (in French) 1995.
[89] I. Podlubny: “Fractional - Order Systems and  Controllers”, IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208-214. 1990.
[90] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, 1999.
[91] I. Podlubuy, L. Dorcak and I. Kostial: “On Fractional Derivatives, Fractional-Order Dynamic Systems and -controllers”, Proc. of the 36th IEEE CDC San Diego, Ca. Dec pp.4985-4900, 1997.
[92] I. Podlubny, “The Laplace Transform Method for Linear Differential Equations of the Fractional Order”, Inst. Exp. Phys., Slovak Acad. Sci., UEF-02-94, Kosice, 1994
[93] I. Podlubny, “Fractional-Order Systems and Fractional-Order Controllers,” Inst. Exp. Phys., Slovak Acad. Sci., UFF-03-94, Kosice, 1994.
[94] M. A. Naimark, “Linear differential operators,” Nauka, Moscow, 1969.
[95] I. Podlubny, “Numerical solution of ordinary fractional differential equations by the fractional difference method,” in: S. Elaydi, I. Gyori and G. Ladas, (eds.), Advances in Difference Equations, Gordon and Breach, Amsterdam, pp.507-516, 1997.
[96] C. Lubich, “Discretized fractional calculus”, SIAM J. Math. Anal., vol.17, no.3, May pp.704-719, 1986.
[97] M. F. Gardner, and J. L.: Barnes: “Transient in Linear Systems” Vol.1, John Wiley & Sons, Inc., New York, 1942.
[98] I. Podlubny, “Fractional differential equation,” Academic Press, 1999.