中 文 摘 要
自然界有許多現象可藉由偏微分方程是來表示如熱傳方程式，波動方程式...等。然而，高階偏微分方程式的解一般不容易求出除非很特殊的例子。偏微分方程式不亦解出的原因乃因為其解如黑盒子般，以人工方式幾乎不可能找出其解。因此，一可逼近非線性方程解的方法必須應用於此。近一二十年來，模糊理論已被證實為很好的逼近法則故其乃是一相當適合用於逼近偏微分方程解的方法。本文試著提出一以適應模糊觀點求解偏微分之法則。此設計之目的乃希望對面臨之偏微分方程式與其對應之邊界條件求得一精確解。藉由模糊邏輯的觀念，吾人可將偏微分方程式之解可先以一包含可調參數（歸屬函數之標準差、推論部分之參數）之粗略模糊解形式加以描述。經由推導，吾人可得一組將上述模糊解精細修正之適應參數調整律。其原理乃藉最佳化之方法將適當之誤差函數極小化而得。其中，實際之偏微分方程式解與提出之模糊解間之逼近誤差的界與歸屬函數數目跟推論部分參數之關係亦於精細的推導之下求出。

此外，偏微分方程式其邊界條件為Dirichlet boundary condition 或 Dirichlet與Neuman之混和型其網格點上之誤差藉參數適應律調整後可證明其隨著時間趨近於無限大將收斂到零。此論文，將藉由幾個工程上常遇到之偏微分方程式來驗證所提之方法並將其結果與有限元素法做比較以證明其成效。

Abstract
It is well-known that a lot of phenomena of nature can be presented by the partial differential equations (PDE’s) or the systems of combination of PDE’s such as heat equations, wave equations and so on. Hence, studies of PDE’s have become one of the main topics of modern mathematical analysis and attracted many people’s attentions. However, the exact solutions for the PDE’s or the systems of PDE’s can not easily be found except for very simple cases or special cases. In recent years, many methods have been developed so far for solving some kinds of PDE’s. For example, Kharab and Kharab describe the use of spreadsheet programs for the numerical solution of the hyperbolic equation [1]. Besides, some produce a solution in the form of an array that contains the value of the solution at a selected group of points [2]. Others use finite element methods that are famous and widely adopted in the mechanical fields to solve some specific partial differential equations [3-5]. In general, a finite element solution only can offer one discrete solution to approximate the exact solution of PDE’s, but this solution often is limited differentiable. Besides, a functional especially for complex cases of PDE’s that can not be easily be derived from the variational calculus should be given first by the finite element method. Furthermore, how close this computed solution is to the exact solution and whether it converges to the exact solution, can not easily checked in practice.

Hence, the investigation of an effective and more correct method for solving the PDE’s or the systems of PDE’s is an important task; it still has no valuable methods to overcome such a solution finding problem currently. As mentioned in the above, one can not easily obtain a solution for PDE’s or the systems of PDE’s unless the complex and tedious mathematical analysis or very simple cases. The main reason why one can not find a solution for the complex PDE’s (high order and nonlinear types) or systems of PDE’s is exact solutions like unknown black boxes, and the profile of the solution is almost not capable of guessing by human works or any mathematical analysis. Since the solution of PDE’s or the systems of PED’s can be regarded as an unknown system, one approximated method that can approach any unknown systems is able to use for finding a suitable and accurate solution for the treated PDE’s or systems of PDE’s. In this thesis, we view the traditional problems of PDE’s mentioned in the above from a different perspective with the help of fuzzy logic systems. Fuzzy logic systems have been widely used in the system modeling to approach the nonlinear unknown systems or control designs in recent years [6-8]. In most of these fuzzy system designs, the fuzzy systems were thought to be a universal approximator [9-12] for any nonlinear systems. Fuzzy logic system has also been proved to be a very good representation for a class of nonlinear dynamic systems by the conventional schemes, and any nonlinear unknown system can be approximated to any desired accuracy as possible. For this reason, an advanced method that relies on the function approximation capabilities of the fuzzy logic systems and results in the construction of a solution written in a differentiable, interpolation form for solving the problems of PDE’s will be proposed in this thesis. This form employs a regression form of fuzzy logic system as the basic approximation element, whose parameters (standard variations and parameters in consequent parts) are adjusted to minimize an appropriate error function. Besides, an elegant proof of the approximated error bound between the exact solution and the proposed fuzzy solution is derived, and this proof can be easily extended to high dimensional cases. Moreover, one sufficient condition for the convergence of the proposed fuzzy solution in the mesh points is offered.

A new technique based on adaptive fuzzy algorithm for the finding of solutions of partial differential equations is presented. The design objective is to find one fuzzy solution to satisfy the encountered partial differential equations and initial/boundary conditions as precise as possible. Based on the concept of fuzzy logic systems, a rough fuzzy solution with adjustable parameters for the partial differential equation is first described. Then, a set of adaptive laws for tuning the free parameters (standard deviations of membership functions) is derived from minimizing an appropriate error function. Besides, an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of membership functions and solution errors has also been proven. Furthermore, the error equations in mesh points are also proven to converge to zero for a class of partial differential equations with one sufficient condition. In this dissertation, we confirm the proposed method by solving a variety of partial differential equations that is practical and encountered in engineering, and present the comparisons with solutions obtained by the finite element method.

References
[1] A. Kharab and R. Kharab, “Spreadsheet solution of hyperbolic partial differential equation,” IEEE Trans. Education, vol. 40, no. 1, pp. 103-110, 1997.
[2] D. Kincaid and W. Cheney, Numerical Analysis. Monterey, CA: Brooks/Cole, 1991.
[3] O. Bi’ro’ and K. Preis, “Finite element analysis of 3-D eddy currents,” IEEE Trans. Magnetics, vol. 26, no. 2, pp. 418-423, 1990.
[4] B. H. Mcdonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory and Techniques, vol. 20, no. 12, pp. 841-847, 1972.
[5] S. Gratkowski, L. Pichon and A. Razek, “New infinite elements for a finite element analysis of 2-D scatting problems,” IEEE Trans. Magnetics, vol. 32, no. 3, pp. 882-885, 1996.
[6] O. H. Wang, K. Tanaka, and F. M. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Trans. Fuzzy Systm., vol. 4, no. 1, pp. 14-23, 1996.
[7] T. Takagi and M. Sugeno, “Fuzzy identification of systems and it applications to modeling and control,” IEEE Trans. Systm. Man, and Cyber., vol. 15, pp. 116-132, 1985.
[8] K. Tanaka, T. Ikeda, and O. H. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, control theory, and linear matrix inequalities,” IEEE Trans. Fuzzy Systm., vol. 4, no. 1, pp. 1-13, 1996.
[9] J.-S. R. Jang, C. T. Sun, and E. Mizutani, Neuro-fuzzy and soft computing. a computational approach to learning and machine intelligence. Upper saddle River, NJ: Prentice-Hall, 1997.
[10] B. Kosko, Fuzzy Sets, Uncertainty and Information. Upper saddle River, NJ: Prentice Hall, 1997.
[11] L. X. Wang, A course in fuzzy systems and control. Upper saddle River, NJ: Prentice Hall, 1997.
[12] L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Englewood Cliffs, NJ: Prentice-Hall, 1994.
[13] S. P. Banks, Control Systems Engineering, NJ: Prentice-Hall, 1986.
[14] D. Kincaid and W. Cheney, Numerical Analysis. Pacific Grove, CA: Brooks/Cole, 1991.
[15] R. Fletcher, Practical Methods of Optimization, 2nd ed. New York: Wiley, 1987.
[16] G. Rudolph, “Convergence analysis of canonical genetic algorithms,” IEEE Trans. Neural Networks, vol. 5, no. 1, pp. 96-101, 1994.
[17] H. G. Kaper and M. Garbey, Asymptotic analysis and the numerical solution of partial differential equations, Marcel Dekker, NJ, 1991.
[18] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th ed., vol.1 New York: McGraw-Hill, 1989.
[19] D. H. Norrie and G. de Viries, An Introduction to Finite Element Analysis, New York: Academic Press, Inc., 1978.
[20] Tom M., Apostol, Mathematical Analysis, Addison Wesley, 1974.
[21] S. S. Saatry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[22] J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall, 1991.
[23] J. M. Ortega, Matrix Theory, NY: Plenum Press, Inc, 1987.