高等應(yīng)用數(shù)學(xué)問題的matlab求解05

1、星期四, 2008-4- 24, 10:46:42,Slide 1 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Chapter 5Integral Transforms and

2、 Complex Variable Functions,Solving Applied Mathematical Problems with MATLAB,CRC/Taylor & Francis PressChinese version by Tsinghua University Press,PPT by Wenbin Dong and Jun Peng, Northeastern University, PRCProo

3、fread by Dingyu Xue & YangQuan Chen,星期四, 2008-4- 24, 10:46:42,Slide 2 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press

4、, 2008,Chapter 5 Integral Transforms and Complex Variable Functions,Laplace Transforms and Their InversesFourier Transforms and Their InversesOther Integral TransformsZ Transforms and Their InversesSolving Complex Va

5、riable Function ProblemsChapter summary,星期四, 2008-4- 24, 10:46:42,Slide 3 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Pres

6、s, 2008,5.1 Laplace Transforms and Their Inverses,Definitions and propertiesComputer solutions to Laplace transform problems,星期四, 2008-4- 24, 10:46:42,Slide 4 (of 102) Dingyü Xue and Ya

7、ngQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.1.1 Definitions and properties,Mathematical definition of the one-sided Laplace transformProperties of Laplace transform

8、:Linear propertywhere and are scalars.,星期四, 2008-4- 24, 10:46:42,Slide 5 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB

9、, CRC Press, 2008,Time-domain shift -domain propertyDifferentiation propertyThe nth order derivative,星期四, 2008-4- 24, 10:46:42,Slide 6 (of 102) Dingyü Xue and YangQuan Chen, Sol

10、ving Applied Mathematical Problems with MATLAB, CRC Press, 2008,when initial values are 0, then, Integration propertyZero initial conditions:the multiple integral:Initial value property,星期四, 2008

11、-4- 24, 10:46:42,Slide 7 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Final value propertyIf has no

12、 pole with non-negative real partConvolution propertywhere the convolution operator is defined as,星期四, 2008-4- 24, 10:46:42,Slide 8 (of 102) Dingyü Xue and YangQuan Chen, Solvin

13、g Applied Mathematical Problems with MATLAB, CRC Press, 2008,Other propertiesInverse Laplace transform: where is greater than the real part of the poles of function,星期四, 2008-4- 24, 10:46:42,S

14、lide 9 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.1.2 Computer solutions to Laplace transform problems,Proble

15、m solution procedures for Laplace transform:Define a symbolic variable such as t , and define time-domain functionDirectly call Laplace() functionor,星期四, 2008-4- 24, 10:46:42,Slide 10 (of 102)

16、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Call pretty() or latex() function to further process the obtained symbolic resultsInverse Laplace

17、 transform:The syntax:default variable isspecify the domain variables and,星期四, 2008-4- 24, 10:46:42,Slide 11 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied

18、Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.1,Given perform its Laplace transformMATLAB solutions:Simplify the answerResult:,星期四, 2008-4- 24, 10:46:42,Slid

19、e 12 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.2,Given obtai

20、n its Laplace and inverse Laplace transformsLaplace transform:Inverse Laplace transform:,星期四, 2008-4- 24, 10:46:42,Slide 13 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied

21、 Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.3,Solve the inverse Laplace transform:Direct solution:High precision numerical solution:,星期四, 2008-4- 24, 10:46:42,Slide 14 (of 102)

22、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.4,Given , derive the relationship

23、between and . and comparison of the difference,星期四, 2008-4- 24, 10:46:42,Slide 15 (of 102) Dingyü Xue and YangQuan Chen,

24、 Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Consider the initial conditions:,星期四, 2008-4- 24, 10:46:42,Slide 16 (of 102) Dingyü Xue and YangQuan Chen,

25、 Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.5,Display the differentiation property of Laplace transformMATLAB solutions:Laplace transform of the eighth-order derivati

26、ve:,星期四, 2008-4- 24, 10:46:42,Slide 17 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.6,For

27、 solveMATLAB solutions: Collect terms in the numerator polynomial:Results,星期四, 2008-4- 24, 10:46:42,Slide 18 (of 102) Dingyü Xue and YangQuan Chen, So

28、lving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.2 Fourier Transforms and Their Inverses,Definitions and propertiesSolving Fourier transform problemsFourier sine and cosine transformsDi

29、screte Fourier sine, cosine transforms,星期四, 2008-4- 24, 10:46:42,Slide 19 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press

30、, 2008,5.2.1 Definitions and properties,Definition of the Fourier transform:Definition of the inverse Fourier transform:,星期四, 2008-4- 24, 10:46:42,Slide 20 (of 102) Dingyü Xue and Yan

31、gQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Properties of the Fourier transform:Linear property: for scalars andShift propertyComplex domain shiftDifferentiation

32、 propertyFourier transform of the nth derivative,星期四, 2008-4- 24, 10:46:42,Slide 21 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLA

33、B, CRC Press, 2008,Integration propertythe Fourier transform to the nth order integralScaling propertyConvolution property,星期四, 2008-4- 24, 10:46:42,Slide 22 (of 102) Dingyü Xu

34、e and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.2.2 Solving Fourier transform problems,The syntax of Fourier transformFourier transformTransform the function o

35、f into a function of,星期四, 2008-4- 24, 10:46:42,Slide 23 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The syn

36、tax of Inverse Fourier transformationInverse Fourier transformTransform the function of into a function of,星期四, 2008-4- 24, 10:46:42,Slide 24 (of 102) Dingyü Xue and YangQuan Chen

37、, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,In MATLAB the definition of Fourier transformIn MATLAB the definition of inverse Fourier transform,星期四, 2008-4- 24, 10:46:42,Slide

38、 25 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.7,Given , where compute

39、the Fourier transform forFourier transform:Inverse Fourier transform:,星期四, 2008-4- 24, 10:46:42,Slide 26 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathemati

40、cal Problems with MATLAB, CRC Press, 2008,Example 5.8,Given with , compute the Fourier transform Simplified and reduced result:,星期四, 2008-4- 24, 10:46:42,Slide 27 (of 102)

41、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.9,Given that , using fourier() command and

42、 the direct integration method respectively to compute the Fourier transform.Using fourier() command:,星期四, 2008-4- 24, 10:46:42,Slide 28 (of 102) Dingyü Xue and YangQuan Chen, Solving

43、Applied Mathematical Problems with MATLAB, CRC Press, 2008,Using direct integration method:NOTE: Not all functions have their corresponding Fourier transforms,星期四, 2008-4- 24, 10:46:42,Slide 29 (of

44、 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.2.3 Fourier sinusoidal and cosine transforms,The Fourier sinusoidal tr

45、ansform is defined asThe Fourier cosine transform is defined as,星期四, 2008-4- 24, 10:46:42,Slide 30 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Pr

46、oblems with MATLAB, CRC Press, 2008,The inverse Fourier sinusoidal transform is defined asThe inverse Fourier cosine transform is defined as,星期四, 2008-4- 24, 10:46:42,Slide 31 (of 102) Din

47、gyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.10,Givencompute the Fourier cosine transformsMATLAB commands:,星期四, 2008-4- 24, 10:46:42,Slide

48、 32 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Fourier sine transform Fourier cosine transform,Calling Maple

49、 functions,星期四, 2008-4- 24, 10:46:42,Slide 33 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,inverse Fourier sine

50、transforminverse Fourier cosine transform,星期四, 2008-4- 24, 10:46:42,Slide 34 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, C

51、RC Press, 2008,Example 5.11,Given , solve its Fourier cosine transform and inverse Fourier cosine transform using Maple functions. Fourier cosine transform:Inverse transfo

52、rm:,星期四, 2008-4- 24, 10:46:42,Slide 35 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.12,Given solve

53、its Fourier cosine transformMATLAB solutions:,星期四, 2008-4- 24, 10:46:42,Slide 36 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, C

54、RC Press, 2008,5.2.4 Discrete Fourier sine, cosine transforms,Discrete Fourier sinusoidal transformDiscrete Fourier cosine transform,星期四, 2008-4- 24, 10:46:42,Slide 37 (of 102) Dingyü

55、; Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The inverse discrete Fourier sinusoidal transformThe inverse discrete Fourier cosine transform,星期四, 2008-4- 2

56、4, 10:46:42,Slide 38 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.13,Givenwhere ,compute t

57、he discrete Fourier sinusoidal transform.MATLAB solutions:,星期四, 2008-4- 24, 10:46:42,Slide 39 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems w

58、ith MATLAB, CRC Press, 2008,5.3 Other Integral Transforms,Mellin transformHankel transform solutions,星期四, 2008-4- 24, 10:46:42,Slide 40 (of 102) Dingyü Xue and YangQuan Chen, Solving

59、Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.3.1 Mellin transform,The Mellin transform is defined asThe inverse Mellin transform is defined as,星期四, 2008-4- 24, 10:46:42,Slide 41 (of 1

60、02) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.14,Given , where , solve its

61、 Mellin transform.MATLAB solutions:Result:,星期四, 2008-4- 24, 10:46:42,Slide 42 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB,

62、CRC Press, 2008,Example 5.15,Given solve its Mellin transforms for several and try to summarize the possible general case. MATLAB solutions for :,星期四,

63、 2008-4- 24, 10:46:42,Slide 43 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,For general case, the Mellin transfo

64、rm equation is:The syntaxMellin transformInverse Mellin transforms,星期四, 2008-4- 24, 10:46:42,Slide 44 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathemat

65、ical Problems with MATLAB, CRC Press, 2008,Example 5.16,Given compute Mellin transform with Maple, then perform inverse Mellin transformMATLAB solutions:,星期四, 2008-4-

66、24, 10:46:42,Slide 45 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,5.3.2 Hankel transform solutions,The th ord

67、er Hankel transform is defined as: where is a Bessel functionThe syntax of evaluating the Hankel transform,星期四, 2008-4- 24, 10:46:42,Slide 46 (of 102) Dingyü Xue and YangQ

68、uan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The th order inverse Hankel transform is defined as:The syntax of evaluating inverse Hankel transform,星期四, 2008-4- 24, 1

69、0:46:42,Slide 47 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 5.17,Given , compute the zerot