微分方程-臺大電信所數(shù)位影像處理與信號處理實驗室

1、1,工程數(shù)學--微分方程,授課者：丁建均,Differential Equations (DE),教學網(wǎng)頁：http://djj.ee.ntu.edu.tw/DE.htm(請上課前來這個網(wǎng)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： 週一至週五的下午皆可來找我  &#

2、160;   個人網(wǎng)頁：http://disp.ee.ntu.edu.tw/          E-mail:  jjding@ntu.edu.tw,上課時間： 星期三 第 3, 4 節(jié) (AM 10:20~12:10)   星期五 第 2 節(jié) (A

3、M 9:10~10:00)上課地點： 電二143課本： "Differential Equations-with Boundary-Value Problem", 8th edition, Dennis G. Zill and Michael R. Cullen, 2016. (metric versi

4、on)評分方式：四次作業(yè)二次小考 15%,   期中考 42.5%,  期末考 42.5%,3,注意事項：請上課前，來這個網(wǎng)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(yè)不可以抄襲。作業(yè)若寫錯但有用心寫仍可以有40%~90% 的分數(shù)，但抄襲或借人抄襲不給分。(4) 我週一至週五下午都在辦公室，

5、有什麼問題 ，歡迎同學們來找我,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),,,Transforms,,,Partial DE (Chap. 12),,,Laplace Transfor

6、m (Chap. 7),,Fourier Series (Chap. 11),,Fourier Transform (Chap. 14),,矩陣解 (Chap. 8，只教不考),6,Chapter 1 Introduction to Differential Equations,1.1 Definitions and Terminology (術(shù)語),Differential Equation (DE): any equation

7、containing derivation (text page 2, definition 1.1) x: independent variable 自變數(shù) y(x): dependent varia

8、ble 應變數(shù),7,Note: In the text book, f(x) is often simplified as f notations of differentiation , , , , ………. Leibniz notation , , ,

9、 , ………. prime notation , , , , ………. dot notation , , , , ………. subscript notation,8,(2) Ordinary Differential

10、 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) Order of a Differentiation Equ

11、ation: the order of the highest derivative in the equation,,7th order,2nd order,10,(5) Linear Differentiation Equation:,,All of the coefficient terms am(x) m = 1, 2, …, n are independent of y.,Property of linear differe

12、ntiation equations: If and y3 = by1 + cy2, then,(if g(x) is treated as the input and y(x) is the output),11,(6) Non-Linear Differentiation Equation,12,(7) Explicit Solution (text page 6) The solution is express

13、ed as y = ?(x)(8) Implicit Solution (text page 7)Example: , Solution: (implicit solution)

14、 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 = x+2

15、 … 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.

16、 and y(0) = 2 y = x+2 and y(2) =3.5 y = x+1.5,,,14,The kth order differential equation usually requires

17、 k initial conditions (or k boundary conditions) to obtain the unique solution. solution: y = x2/2 + bx + c, b and

18、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 ~ (k–1)th derivatives at some

19、 points.,(boundary conditions，在不同點),(boundary conditions，在不同點),(initial conditions ，在相同點),15,1.3 Differential Equations as Mathematical Model,Physical meaning of differentiation: the variation at certain time or certai

20、n place,Example 1:,,x(t): location, v(t): velocity, a(t): accelerationF: force, β: coefficient of friction, m: mass,16,A: population人口增加量和人口呈正比,Example 2: 人口隨著時間而增加的模型,17,T: 熱開水溫度, Tm: 環(huán)境溫度t: 時間,Example

21、 3: 開水溫度隨著時間會變冷的模型,18,大一微積分所學的：,的解,,問題：,(1) 若等號兩邊都出現(xiàn) dependent variable (如 pages 16, 17 的例子),(2) 若order of DE 大於 1,例如：,該如何解？,19,Review dependent variable and independent variable DE PDE and ODE Order of DE linear

22、DE and nonlinear DE explicit solution and implicit solution initial value; boundary value IVP,20,Chapter 2 First Order Differential Equation,2-1 Solution Curves without a Solution,Instead of using analytic methods

23、, 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),,21,Example 1 dy/dx = 0.2xy,From： Fig. 2-1-3(a) in “Differential Equations-with Boundary-Va

24、lue Problem”, 8th ed., Dennis G. Zill and Michael R. Cullen.,22,From： Fig. 2-1-4 in “Differential Equations-with Boundary-Value Problem”, 8th ed., Dennis G. Zill and Michael R. Cullen.,Example 2 dy/dx = sin(y),

25、 y(0) = –3/2 With initial conditions, one curve can be obtained,23,Advantage: It can solve some 1st order DEs that cannot be solved by mathematics.Disadvantage:It can only be used for the case of the

26、 1st order DE.It requires a lot of time,24,Section 2-6 A Numerical Method,Another way to solve the DE without analytic methods independent variable x x0, x1, x2, ………… Find the

27、solution of Since approximation,,sampling(取樣),,前一點的值,,,,取樣間格,25,Example: dy(x)/dx = 0.2xy y(xn+1) = y(xn) + 0.2xn y(xn )*(xn+1 –xn).dy/dx = sin(x)

28、 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)

29、的例子,Constraint for obtaining accurate results: (1) small sampling interval (2) small variation of f(x, y),26,(a),(b),(c),(d),27,,Advantages -- It can solve some 1st order DEs that cannot be solved by mathemat

30、ics.-- can be used for solving a complicated DE (not constrained for the 1st order case) -- suitable for computer simulation Disadvantages -- numerical error (數(shù)值方法的課程對此有詳細探討),28,Exercises for Practicing (n