工程數(shù)學--微分方程

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： 星期三下午 1:00~5:00  

2、    個人網(wǎng)頁：http://disp.ee.ntu.edu.tw/          E-mail:  djj@cc.ee.ntu.edu.tw,    djj1@ms63.hinet.net,上課時間： 星期三 第 3, 4 節(jié) (AM 10:20~12:10) 

3、60; 星期五 第 2 節(jié) (AM 9:10~10:00)上課地點： 電二143課本： "Differential Equations-with Boundary-Value Problem", 7th edition, Dennis G. Zill and Michael R. Cullen評分方式：

4、四次作業(yè)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(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) 我週一至週四下午都在辦公室，有什麼問題 ，歡迎同學們來

5、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),,,,矩陣解法 (Chap. 8),Transforms,,,Partial DE (Chap. 12),,,Laplace Trans

6、form (Chap. 7),,Fourier Series (Chap. 11),,Fourier Transform (Chap. 14),6,Chapter 1 Introduction to Differential Equations,1.1 Definitions and Terminology (術語),Differential Equation (DE): any equation containing deriva

7、tion (page 2, definition 1.1) x: independent variable 自變數(shù) y(x): dependent variable 應變數(shù),7,In the text

8、book f(x) is often simplified as f notations of differentiation , , , , ………. Leibniz notation , , , , ………. prime not

9、ation , , , , ………. dot notation , , , , ………. subscript notation,8,(2) Ordinary Differential Equation (ODE): differenti

10、ation 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 Equation: the order of the highe

11、st 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,1

12、1,(6) Non-Linear Differentiation Equation,12,(7) Explicit Solution (page 6) The solution is expressed as y = ?(x)(8) Implicit Solution (page 7)Example: ,

13、 Solution: (implicit solution) or (explicit solution),13,1.2 Initial Value Problem (IVP),A differentiation

14、equation always has more than one solution. for , y = x, y = x+1 , y = x+2 … are all the solutions of the above differentiation equation.General form of the solution: y = x+ c, where c is

15、any constant. The initial value (未必在 x = 0) is helpful for obtain the unique solution. and y(0) = 2 y = x+2

16、 and y(2) =3.5 y = x+1.5,,,14,The kth order differential equation usually requires k initial conditions (or k boundary conditions) to obtain the unique solution.

17、 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) =2

18、For the kth order differential equation, the initial conditions can be 0th ~ (k–1)th derivatives at some points.,(boundary conditions，在不同點),(boundary conditions，在不同點),(initial conditions),15,1.3 Differential Equations

19、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(huán)境溫度t: 時間,Example 2:,17,大一微積分所學的：,的解,,問題：,(1) 若等號兩邊都出現(xiàn) de

20、pendent 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

21、 initial value IVP,19,Chapter 2 First Order Differential 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-axi

22、s,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”, 7th ed., Dennis G. Zill and Michael R. Cullen.,21,資料來源： Fig. 2-1

23、-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 obt

24、ained,22,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 1st order DE.It requires a lot of time,23,Section 2-6 A Numerical Method

25、,Another way to solve the DE without analytic methods independent variable x x0, x1, x2, ………… Find the solution of Since approximation,,sampling(取

26、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),

27、 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 accurate results: (1) small sampling

28、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

29、-- more time for computation -- numerical error (數(shù)值方法的課程對此有詳細探討),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

30、, 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 for linear equation,,,method for exact eq

31、uation,homogeneous equation method,transform,,,,,Laplace transform,Fourier transform,,direct integration,,series solution,,,Bernoulli’s equation method,method for Ax + By + c,,Fourier series,Fourier sine series,,,Fourier

32、 cosine series,29,Simplest method for solving the 1st order DE: Direct Integration dy(x)/dx = f(x) where,30,Table of Integration,,31,2-2 Separable Variables,2-2-1 方

33、法的限制條件,1st order DE 的一般型態(tài): 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.,

34、 dy(x)/dx = g(x)h(y)then the 1st order DE is separable (or have separable variable).,,32,dy(x)/dx = g(x)h(y),條件：,33,If , thenStep 1

35、 where p(y) = 1/h(y)Step 2 where,,,,,,,,,,,2-2-2 解法,(b) Check the singular solution,分

36、離變數(shù),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 for r,,See whether the solution is a special case of the general solution.,35,Exampl

37、e 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,Exam

38、ple 練習小技巧遮住解答和筆記，自行重新算一次(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(jīng)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)

39、 , y(4) = –3,,check the singular solution,,,,(implicit solution),,(explicit solution),valid,invalid,,Step 1,Step 2,Extra Step (a),,Extra Step (b),38,Example 3 (with singular solution, text pag

40、e 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 o

41、f page 48,, y(0) = 0,Step 1,Step 2,Extra Step (a),Extra Step (b) Check the singular solution,,,,,Solution: or,其實，還有更多的解,41,, y(0) = 0,solutions: (1) (2)

42、 (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 x0?h < x < x0 +h 的區(qū)間當中有唯一解,hint: 用「圖解」的角度來思考,43,(1) (2)

43、If dy/dx = g(x) and y(x0) = y0, then,積分 (integral, antiderivative) 難以計算的 function，被稱作是 nonelementary,如 ,,此時，solution 就可以寫成 的型態(tài),2-2-5 Solutions Defined by Integral,44,So

44、lution,Example 5,或者可以表示成 complementary error function,45,? error function (useful in probability),? complementary error function,See text page 59 in Section 2.3,46,(1) 複習並背熟幾個重要公式的積分(2) 別忘了加 c 並且熟悉什麼情況下 c 可以合併和簡

45、化 (3) 若時間允許，別忘了計算 singular solution(4) 多練習，加快運算速度,2-2-6 本節(jié)要注意的地方,47,http://integrals.wolfram.com/index.jsp,附錄二 微分方程查詢,輸入數(shù)學式，就可以查到積分的結果,範例：,(a) 先到integrals.wolfram.com/index.jsp 這個網(wǎng)站,(b) 在右方的空格中輸入數(shù)學式，例如,,,數(shù)學式,48,(c)

46、 接著按 “Compute Online with Mathematica” 就可以算出積分的結果,,,按,,結果,,49,(d) 有時，對於一些較複雜的數(shù)學式，下方還有連結，點進去就可以看到相關的解說,,,連結,50,http://mathworld.wolfram.com/,對微分方程的定理和名詞作介紹的百科網(wǎng)站,http://www.sosmath.com/tables/tables.html,眾多數(shù)學式的 mathem

47、atical table (不限於微分方程),http://www.seminaire-sherbrooke.qc.ca/math/Pierre/Tables.pdf,眾多數(shù)學式的 mathematical table，包括 convolution, Fourier transform, Laplace transform, Z transform,其他有用的網(wǎng)站,軟體當中， Maple, Mathematica, Matlab 皆有微

48、積分結果查詢的功能,51,2-3 Linear Equations,[Definition 2.3.1] The first-order DE is a linear equation if it has the following form: g(x) = 0: homogeneous g(x) ? 0: nonhomogeneous,,“friendly” form of DEs,2-3-1 方法的適用條件,52,

49、,Standard form:,許多自然界的現(xiàn)象，皆可以表示成 linear first order DE,53,2-3-2 解法的推導,,,子問題 1,子問題 2,Find the general solution yc(x)(homogeneous solution),Find any solution yp(x)(particular solution),,,Solution of the DE,54,? yc + yp i

50、s a solution of the linear first order DE, since,Any solution of the linear first order DE should have the form yc + yp . The proof is as follows. If y is a solution of the DE, then,Thus, y ? yp should be the solution

51、 of,y should have the form of y = yc + yp,55,Solving the homogeneous solution yc(x) (子問題一）,,separable variable,,,Set , then,,56,Solving the particular solution yp(x) (子問題二）Set yp(x) = u(x)

52、y1(x) (猜測 particular solution 和 homogeneous solution 有類似的關係),,equal to zero,,,,,,57,,,solution of the linear 1st order DE: where c is any constant : integrating f

53、actor,,58,(Step 1) Obtain the standard form and find P(x) (Step 2) Calculate (Step 3) The standard form of the linear 1st order DE can be rewritten as: (Step 4) Integrate both sides of the above equation,,remember i

54、t,,or remember it, skip Step 3,(Extra Step) (a) Initial value (c) Check the Singular Point,2-3-3 解法,59,Singular points: the locations where a1(x) = 0 i.e., P(x) ? ?More g

55、enerally, even if a1(x) ? 0 but P(x) ? ? or f(x) ? ?, then the location is also treated as a singular point.(a) Sometimes, the solution may not be defined on the interval including the singular points. (such as Example

56、4)(b) Sometimes the solution can be defined at the singular points, such as Example 3,60,More generally, even if a1(x) ? 0 but P(x) ? ? or f(x) ? ?, then the location is also treated as a singular point.,Exercise 29,61

57、,Example 2 (text page 55),,,為何在此時可以將–3x+c 簡化成 –3x?,,,,,check the singular point,2-3-4 例子,Extra Step (c),Step 1,Step 2,Step 3,Step 4,或著，跳過 Step 3，直接代公式,62,Example 3 (text page 56),,Step 1,,Step 2,Step 3,,,Extra Step (

58、c),check the singular point,若只考慮 x > 0 的情形,,x = 0,Step 4,x 的範圍: (0, ?),考慮 x < 0 的情形,修正： x ? (??, ?),63,Example 4 (text page 57),,,check the singular point,,,,defined for x ? (–?, –3), (–3, 3), (3, ?)not include

59、s the points of x = –3, 3,Extra Step (c),64,Example 6 (text, page 58),,,check the singular point,,,0 ? x ? 1,x > 1,,,,,,from initial condition,,,,要求 y(x) 在 x = 1 的地方為 continuous,65,2-3-5 名詞和定義,(1) transient term, st

60、able term,Example 5 (text page 58) 的解為 : transient term 當 x 很大時會消失x ?1: stable term,y,x?1,x-axis,66,(2) piecewise continuous A function g(x) is piecewise continuous in the region of [x1, x2] if g'(x)

61、 exists for any x ? [x1, x2].,In Example 6, f(x) is piecewise 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 fun

62、ction,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: Try to calculate,2-3-6 小技巧,Example:,(not linear, not separable)

63、,,(linear),,(implicit solution),69,2-3-6 本節(jié)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(shù)項可以忽略,70

64、,最上策： realize + remember it上策： realize 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 Equat

65、ions,應用題,Convert a question into a 1st order DE. 將問題翻譯成數(shù)學式 (2) Many of the DEs can be solved by Separable variable method or Linear equation method (with integration table remembra

66、nce),72,3-1 Linear Models,Growth and Decay (Examples 1~3)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 ?

67、 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,Example 1 (an example of growth and decay, text page 83) Initial: A culture (培養(yǎng)皿) initially has P0 number of bacteria. The other initi

68、al 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

69、 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)