1 / 53

化工應用數學

化工應用數學. Series Solutions of Ordinary Differential Equations. 授課教師： 林佳璋. Infinite Series. They can be accepted as solutions if they are convergent . As n  , S n S (some finite number), the series is “ convergent ”. As n  , S n  ±, the series is “ divergent ”.

floyd
Download Presentation

化工應用數學

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 化工應用數學 Series Solutions of Ordinary Differential Equations 授課教師： 林佳璋

  2. Infinite Series • They can be accepted as solutions if they are convergent. • As n, SnS (some finite number), the series is “convergent”. • As n, Sn ±, the series is “divergent”. • In other cases, the series is “oscillatory”. convergent divergent if z is real and positive; oscillatory for all other z divergent oscillatory

  3. Properties of Infinite Series • If a series contains only positive real numbers or zero, it must be either convergent or divergent. • If a series is convergent, then un 0, as n  . • If a series is absolutely convergent, then it is also convergent. • -If the series is convergent, it is • absolutely convergent. Comparison Test N is some finite integer convergent absolutely convergent divergent divergent to 

  4. Comparison Test of Infinite Series (a) p>1 < < convergent (b) p<1 divergent

  5. Comparison Test of Infinite Series (c) p=1 divergent

  6. Ratio Test of Infinite Series convergent divergent but k value is not found, the test is inconclusive

  7. Power Series Consider the power series convergent divergent convergent convergent if z=0

  8. Power Series binomial series z may be complex, but p is real convergent for exponential series convergent

  9. Power Series logarithmic series convergent for trigonometric series convergent

  10. Taylor’s Theorem and L’Hopital’s Rule Taylor’s Theorem If x is a real variable, and f(x) can be differentiated n times at x=a, then 0 as h0 L’Hopital’s Rule Where f(x) and g(x) are differentiable n times, and f(a)=g(a)=0

  11. Example

  12. O.D.E with Analytic Coefficients Consider the O.D.E Where the leading coefficient in some interval of interest ,so that the equation may be written in the form It is assumed that the coefficients P(x), Q(x) and R(x) are analytic functions at x=x0. The solution of the above equation is convergent for

  13. Example Solve Consider x0=0 replacing n by n+2

  14. Example The coefficient of each power of x must be zero; that is,

  15. Example Solve Consider x0=0

  16. Example The coefficient of each power of x must be zero; that is,

  17. Example Solve

  18. Example

  19. Method of Frobenius having a radius of convergence R The equation can be solved by method of Frobenius in the form of a power series which is also convergent for

  20. Method of Frobenius xc indicial equation xc+1

  21. Method of Frobenius Case I. Roots of indicial equation different, but not by an integer xc-1 xc xc+1 xc+r

  22. Method of Frobenius c=0

  23. Method of Frobenius c=-1/2

  24. Method of Frobenius Case II. Roots of indicial equation equal

  25. Method of Frobenius xc-1 xc xc+r

  26. Method of Frobenius c=0

  27. Method of Frobenius

  28. Method of Frobenius Case IIIa. Roots of indicial equation differing by an integer c=0 c=-1

  29. Method of Frobenius Case IIIb. Roots of indicial equation differing by an integer c=2 c=0

  30. Method of Frobenius Example The shape of the cooling fin is illustrated in Fig where the radius of the pipe (a) is 8 cm, the radius of the rim of the fin (b) is 20 cm, and the coordinate x m is measured inwards from the rim of the fin. There are two natural origins for the coordinate, but since the temperature distribution in the vicinity of the pipe axis is of no interest, the origin is taken on the rim instead. Assuming that the fin is thin, temperature variations normal to the central plane of the fin will be neglected. The thermal conductivity of the fin (k) is 380 W/mC, and the surface heat transfer coefficient (h) is 12 W/m2C. Denoting temperature by TC with TAC representing the air temperature, the heat balance can be taken as follows. x b a 

  31. Method of Frobenius Example The area available for heat conduction is

  32. Method of Frobenius Example xc-1 xc xc+r

  33. Method of Frobenius Example The rim temperature is 79.6+16=95.6C (x=0)

  34. Method of Frobenius Example A supply of hot air is to be obtained by drawing cool air through a heated cylindrical pipe. The pipe is 0.1 m diameter and 1.2 m long, and is maintained at a temperature of Tw=300 C throughout its length. The properties of the air are: Heat capacity (Cp) = 1000 J/kg C Thermal conductivity (k) = 0.035 W/m C Density () = 0.8 kg/m3 Flow rate (u) = 0.009 m3/s Inlet temperature = 20 C Overall heat transfer coefficient (h) =10x-1/2 W/m2 C Assuming that heat transfer takes place by conduction within the gas in an axial direction, mass flow of the gas in an axial direction, and by the above variable heat transfer coefficient from the walls of the tube, find the temperature of the exit gas.

  35. Method of Frobenius Example h Tw u Cp 20C T T+T x x Input Output By conduction By mass flow Wall heat transfer

  36. Method of Frobenius Example

  37. Method of Frobenius Example B.C x=0, t=300-20=280 C=280 The exit gas temperature is 192C (x=1.2)

  38. Bessel’s Equation Bessel’s equation of order k, where k is a positive or zero constant. xc xc+1 xc+r

  39. Bessel’s Equation c=k (k+1) is the gamma function Case I (2k is not an integer or zero)

  40. Bessel’s Equation Case II (k=0)

  41. Bessel’s Equation Case IIIa (k is an integer)

  42. Bessel’s Equation Case IIIb (2k is an odd integer) General solution k is not an integer or zero k is an integer or zero

  43. Modified Bessel’s Equation k is not an integer or zero k is an integer or zero k is not an integer or zero k is an integer or zero

  44. r +dr r 1.25 cm 5cm Bessel’s Equation Example Two thin wall metal pipes of 2.5 cm external diameter and joined by flanges 1.25 cm thick and 10 cm diameter, are carrying steam at 120 C. If the conductivity of the flange metal k=400 W/m C and the exposed surfaces of the flanges lose heat to the surrounding at T1=15 C according to a heat transfer coefficient h=12 W/m2C, find the rate of heat loss from the pipe, and the proportion which leaves the rim of the flange.

  45. Bessel’s Equation Example A=103, B=0.477

  46. Properties of Bessel’s Function Behavior Near the Origin Differential Properties

  47. Properties of Bessel’s Function Integral Properties

  48. Properties of Bessel’s Function Negative Integer Order Half Integer Order

  49. Error Function It occurs in the theory of probability, distribution of residence times, conduction of heat, and diffusion matter: erf x z: dummy variable Proof in next slide z 0 x

  50. Error Function x and y are two independent Cartesian coordinates in polar coordinates Error between the volume determined by x-y and r- The volume of  has a base area which is less than 1/2R2 and a maximum height of e-R2

More Related