工程數學--微分方程.ppt

1、1,工程數學-微分方程,授課者：丁建均,Differential Equations (DE),教學網頁：http:/djj.ee.ntu.edu.tw/DE.htm(請上課前來這個網站將講義印好)歡迎大家來修課！,2,授課者：丁建均Office： 明達館723室, TEL： 33669652 Office hour： 星期三下午 1:005:00個人網頁：http:/disp.ee.ntu.edu.tw/ E-mail: djjcc.ee.ntu.edu.tw, ,上課時間： 星期三 第 3, 4 節(AM 10:2012:10) 星期五 第 2 節 (AM 9:1010:00)上課地點：

2、電二143課本： Differential Equations-with Boundary-Value Problem, 7th edition, Dennis G. Zill and Michael R. Cullen評分方式：四次作業一次小考 10%, 期中考 45%, 期末考 45%,3,注意事項：請上課前，來這個網頁，將上課資料印好。 http:/djj.ee.ntu.edu.tw/DE.htm (2) 請各位同學踴躍出席 。(3) 作業不可以抄襲。作業若寫錯但有用心寫仍可以有40%90% 的分數，但抄襲或借人抄襲不給分。(4) 我週一至週四下午都在辦公室，有什麼問題 ，歡迎同學們來找

3、我,4,上課日期,5,課程大綱,Introduction (Chap. 1),First Order DE,Higher Order DE,解法 (Chap. 2),應用 (Chap. 3),解法 (Chap. 4),應用 (Chap. 5),多項式解法 (Chap. 6),矩陣解法 (Chap. 8),Transforms,Partial DE (Chap. 12),Laplace Transform (Chap. 7),Fourier Series (Chap. 11),Fourier Transform (Chap. 14),6,Chapter 1 Introduction to Dif

4、ferential Equations,1.1 Definitions and Terminology (術語),Differential Equation (DE): any equation containing derivation (page 2, definition 1.1) x: independent variable 自變數 y(x): dependent variable 應變數,7,In the text book f(x) is often simplified as f notations of differentiation , , , , . Leibniz no

5、tation , , , , . prime notation , , , , . dot notation , , , , . subscript notation,8,(2) Ordinary Differential Equation (ODE): differentiation with respect to one independent variable,(3) Partial Differential Equation (PDE): differentiation with respect to two or more independent variables,9,(4) Or

6、der of a Differentiation Equation: the order of the highest derivative in the equation,7th order,2nd order,10,(5) Linear Differentiation Equation:,All the coefficient terms are independent of y.,Property of linear differentiation equations: If and y3 = by1 + cy2, then,11,(6) Non-Linear Differentiati

7、on Equation,12,(7) Explicit Solution (page 6) The solution is expressed as y = (x)(8) Implicit Solution (page 7)Example: , Solution: (implicit solution) or (explicit solution),13,1.2 Initial Value Problem (IVP),A differentiation equation always has more than one solution. for , y = x, y = x+1 , y =

8、x+2 are all the solutions of the above differentiation equation.General form of the solution: y = x+ c, where c is any constant. The initial value (未必在 x = 0) is helpful for obtain the unique solution. and y(0) = 2 y = x+2 and y(2) =3.5 y = x+1.5,14,The kth order differential equation usually requir

9、es k initial conditions (or k boundary conditions) to obtain the unique solution. solution: y = x2/2 + bx + c, b and c can be any constant y(1) = 2 and y(2) = 3 y(0) = 1 and y(0) =5 y(0) = 1 and y(3) =2For the kth order differential equation, the initial conditions can be 0th (k1)th derivatives at s

10、ome points.,(boundary conditions，在不同點),(boundary conditions，在不同點),(initial conditions),15,1.3 Differential Equations as Mathematical Model,Physical meaning of differentiation: the variation at certain time or certain place,A: population人口增加量和人口呈正比,Example 1:,16,T: 熱開水溫度, Tm: 環境溫度t: 時間,Example 2:,17,

11、大一微積分所學的：,的解,問題：,(1) 若等號兩邊都出現 dependent variable (如 pages 15, 16 的例子),(2) 若order of DE 大於 1,例如：,18,Review dependent variable and independent variable DE PDE and ODE Order of DE linear DE and nonlinear DE explicit solution and implicit solution initial value IVP,19,Chapter 2 First Order Differential

12、Equation,2-1 Solution Curves without a Solution,Instead of using analytic methods, the DE can be solved by graphs (圖解),slopes and the field directions:,x-axis,y-axis,(x0, y0),the slope is f(x0, y0),20,Example 1 dy/dx = 0.2xy,資料來源： Fig. 2-1-3(a) of “Differential Equations-with Boundary-Value Problem”

13、, 7th ed., Dennis G. Zill and Michael R. Cullen.,21,資料來源： Fig. 2-1-4 of “Differential Equations-with Boundary-Value Problem”, 7th ed., Dennis G. Zill and Michael R. Cullen.,Example 2 dy/dx = sin(y), y(0) = 3/2 With initial conditions, one curve can be obtained,22,Advantage: It can solve some 1st ord

14、er DEs that cannot be solved by mathematics.Disadvantage:It can only be used for the case of the 1st order DE.It requires a lot of time,23,Section 2-6 A Numerical Method,Another way to solve the DE without analytic methods independent variable x x0, x1, x2, Find the solution of Since approximation,s

15、ampling(取樣),前一點的值,取樣間格,24,Example: dy(x)/dx = 0.2xy y(xn+1) = y(xn) + 0.2xn y(xn )*(xn+1 xn).dy/dx = sin(x) y(xn+1) = y(xn) + sin(xn)*(xn+1 xn). .,後頁為 dy/dx = sin(x), y(0) = 1,(a) xn+1 xn = 0.01, (b) xn+1 xn = 0.1, (c) xn+1 xn = 1, (d) xn+1 xn = 0.1, dy/dx = 10sin(10x) 的例子,Constraint for obtaining a

16、ccurate results: (1) small sampling interval (2) small variation of f(x, y),25,(a),(b),(c),(d),26,Advantages - can be used for solving a complicated DE (not constrain for the 1st order case) - suitable for computer simulation Disadvantages - more time for computation - numerical error (數值方法的課程對此有詳細探

17、討),27,Exercises for Practicing (not homework, but are encouraged to practice)1-1: 1, 13, 19, 23, 331-2: 3, 13, 21, 331-3: 2, 7, 282-1: 1, 13, 20, 25, 332-6: 1, 3,28,附錄一 Methods of Solving the First Order Differential Equation,graphic method,numerical method,analytic method,separable variable,method

18、for linear equation,method for exact equation,homogeneous equation method,transform,Laplace transform,Fourier transform,direct integration,series solution,Bernoullis equation method,method for Ax + By + c,Fourier series,Fourier sine series,Fourier cosine series,29,Simplest method for solving the 1st

19、 order DE: Direct Integration dy(x)/dx = f(x) where,30,Table of Integration,31,2-2 Separable Variables,2-2-1 方法的限制條件,1st order DE 的一般型態: dy(x)/dx = f(x, y),Definition 2.2.1 (text page 45) If dy(x)/dx = f(x, y) and f(x, y) can be separate as f(x, y) = g(x)h(y) i.e., dy(x)/dx = g(x)h(y)then the 1st or

20、der DE is separable (or have separable variable).,32,dy(x)/dx = g(x)h(y),條件：,33,If , thenStep 1 where p(y) = 1/h(y)Step 2 where,2-2-2 解法,(b) Check the singular solution,分離變數,個別積分,Extra Step: (a) Initial conditions,34,Extra Step (b) Check the singular solution:,Suppose that y is a constant r,solution

21、 for r,See whether the solution is a special case of the general solution.,35,Example 1 (text page 46) (1 + x) dy y dx = 0,check the singular solution,set y = r , 0 = r/(1+x) r = 0, y = 0,2-2-3 Examples,(a special case of the general solution),Extra Step (b),Step 1,Step 2,36,Example 練習小技巧遮住解答和筆記，自行重

22、新算一次(任何和解題有關的提示皆遮住)Exercise 練習小技巧初學者，先針對有解答的題目作練習累積一定的程度和經驗後，再多練習沒有解答的題目將題目依類型分類，多綀習解題正確率較低的題型動筆自己算，就對了,37,Example 2 (with initial condition and implicit solution, text page 46) , y(4) = 3,check the singular solution,(implicit solution),(explicit solution),valid,invalid,Step 1,Step 2,Extra Step (a),

23、Extra Step (b),38,Example 3 (with singular solution, text page 47),check the singular solution,set y = r , 0 = r2 4 r = 2, y = 2,Extra Step (b),Step 1,Step 2,or,y = 2,39,Example 4 (text page 47)自修注意如何計算 ,40,Example in the bottom of page 48, y(0) = 0,Step 1,Step 2,Extra Step (a),Extra Step (b) Check

24、the singular solution,Solution: or,其實，還有更多的解,41, y(0) = 0,solutions: (1) (2) (3),b 0 a,42,2-2-4 IVP 是否有唯一解？,這個問題有唯一解的條件：(Theorem 1.2.1, text page 15),如果 f(x, y), 在 x = x0, y = y0 的地方為 continuous,則必定存在一個 h，使得 IVP 在 x0h x 0 的情形,x = 0,Step 4,x 的範圍: (0, ),考慮 x 1,from initial condition,要求 y(x) 在 x = 1 的地

25、方為 continuous,65,2-3-5 名詞和定義,(1) transient term, stable term,Example 5 (text page 58) 的解為 : transient term 當 x 很大時會消失x 1: stable term,y,x1,x-axis,66,(2) piecewise continuous A function g(x) is piecewise continuous in the region of x1, x2 if g(x) exists for any x x1, x2.,In Example 6, f(x) is piecewi

26、se continuous in the region of 0, 1) or (1, ),(3) Integral (積分) 有時又被稱作 antiderivative,(4) error function,complementary error function,67,(5) sine integral function,Fresnel integral function,(6),f(x) 常被稱作 input 或 deriving function,Solution y(x) 常被稱作 output 或 response,68,When is not easy to calculate:

27、 Try to calculate,2-3-6 小技巧,Example:,(not linear, not separable),(linear),(implicit solution),69,2-3-6 本節要注意的地方,(1) 要先將 linear 1st order DE 變成 standard form(2) 別忘了 singular point,注意：singular point 和 Section 2-2 提到的 singular solution 不同,(3) 記熟公式,或,(4) 計算時， 的常數項可以忽略,70,最上策： realize + remember it上策： re

28、alize it中策： remember it 下策： read it without realization and remembrance最下策： rest z.z.z,太多公式和算法，怎麼辦？,71,Chapter 3 Modeling with First-Order Differential Equations,應用題,Convert a question into a 1st order DE. 將問題翻譯成數學式 (2) Many of the DEs can be solved by Separable variable method or Linear equation me

29、thod (with integration table remembrance),72,3-1 Linear Models,Growth and Decay (Examples 13)Change the Temperature (Example 4)Mixtures (Example 5)Series Circuit (Example 6),可以用 Section 2-3 的方法來解,73,翻譯 A(0) = P0,翻譯 A(1) = 3P0/2,翻譯 k is a constant,這裡將課本的 P(t) 改成 A(t),翻譯 find t such that A(t) = 3P0,Ex

30、ample 1 (an example of growth and decay, text page 83) Initial: A culture (培養皿) initially has P0 number of bacteria. The other initial condition: At t = 1 h, the number of bacteria is measured to be 3P0/2. 關鍵句: If the rate of growth is proportional to the number of bacteria A(t) presented at time t, Question: determine the time necessary for the number of bacteria to triple,74,A(0) = P0, A(1) = 3P0/2,可以用 什麼方法解？,check singular solution,Step 1,Step 2,Extra Step (b),Extra Step (a),(1) c = P0 (2) k = ln(3/2) = 0.4055,