1 / 40

Chapter 5 (5.4~5.6)

Kreyszig, E. “Advanced Engineering Mathematics” (9 th ed. ). Chapter 5 (5.4~5.6). ◎ Series Solutions of ODEs. 微分方程式之級數解 ◎ Special Functions 特殊方程式. Contents. 5.4 Frobenius Method 5.5 Bessel ’ s Equation. Bessel Functions J ν ( x ) 5.6 Bessel Functions of the Second Kind Y ν ( x ).

waseem
Download Presentation

Chapter 5 (5.4~5.6)

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. Kreyszig, E. “Advanced Engineering Mathematics” (9th ed. ) Chapter 5(5.4~5.6) ◎Series Solutions of ODEs. 微分方程式之級數解 ◎Special Functions 特殊方程式

  2. Contents • 5.4 Frobenius Method • 5.5 Bessel’s Equation. Bessel Functions Jν(x) • 5.6 Bessel Functions of the Second Kind Yν(x)

  3. THEOREM 1 Frobenius Method (傅洛貝尼斯法) Let b(x) and c(x) be any functions that are analytic at x = 0. Then the ODE (1) has at least one solution that can be represented in the form (2) where the exponent r may be any (real or complex) number (and r is chosen so that a0≠ 0). 5.4 Power Frobenius Method(1/20)

  4. 5.4 Power FrobeniusMethod(2/20) We shall now explain the Frobenius method for solving (1). Multiplication of (1) by x2 gives the more convenient form (1')x2y" + xb(x)y' + c(x)y = 0. We first expand b(x) and c(x) in power series, b(x) = b0 + b1x +b2x2 + ‥‥ , c(x) = c0 + c1x +c2x2 + ‥‥ or we do nothing if b(x) and c(x) are polynomials.

  5. 5.4 Power FrobeniusMethod(3/20) Then we differentiate (2) term by term, finding By inserting all these series into (1') we readily obtain xr[r(r – 1)a0 + ‥‥] + (b0 + b1x + ‥‥)xr(ra0 + ‥‥) + (c0 + c1x + ‥‥)xr(a0 + a1x + ‥‥) = 0

  6. 5.4 Power FrobeniusMethod(4/20) We now equate the sum of the coefficients of each power xr, xr+1, xr+2, ‥‥ to zero. The equation corresponding to the power xris [r(r – 1) + b0r + c0]a0 = 0 Since by assumption a0≠ 0, the expression in the brackets [‥‥] must be zero. This gives (4) r(r – 1) + b0r + c0 = 0 This important quadratic equation is called the indicial equation (指示方程式).

  7. THEOREM 2 Frobenius Method. Basis of Solutions. Three Cases(1) Suppose that the ODE (1) satisfies the assumptions in Theorem 1. Let r1 and r2 be the roots of the indicial equation (4). Then we have the following three cases. Case 1. Distinct Roots Not Differing by an Integer. A basis is (5) and (6) 5.4 Power FrobeniusMethod(5/20)

  8. 5.4 Power FrobeniusMethod(6/20) • Example (A) : Solve 4xy”+2y’ +y= 0 by Frobenius method with coefficients represented up to and including x3？ Since & 代入ODE   **取最小冪次項 xr1 ( i.e. m=0)之係數為零 indicial eq. 4r(r1)+2r=0  r(2r1) =0  r =0 or r=1/2 (case 1.)

  9. 5.4 Power FrobeniusMethod(7/20) (when r=0) 級數式 註標變換 (第1,2項 m1=s, m=s+1；第3項 m=s)  [4(s+1)s+2(s+1)]as+1 + as=0  recurrence a1=  a0/2! ; a2=  a1/12=a0/4! ; a3=  a2/30=  a0/6! … 取a0=1  ＃

  10. 5.4 Power FrobeniusMethod(8/20) (when r=1/2) 級數式 註標變換 (第1,2項 m1/2=s, m=s+1/2；第3項 m+1/2=s, m=s1/2)  [4(s+1)s+2(s+1)]as+1 + as=0  recurrence a1=  a0/3!(s=1/2); a2=  a1/20=a0/5!(s=3/2); a3=  a2/42=a0/7!(s=5/2); … 取a0=1  ＃

  11. THEOREM 2 Frobenius Method. Basis of Solutions. Three Cases(2) Case 2. Double Root r1= r2= r. A basis is (7) (of the same general form as before) and (8) 5.4 Power FrobeniusMethod(9/20)

  12. 5.4 Power FrobeniusMethod(10/20) • Example (B) : Solve (x2x)y”+(3x1)y’ +y= 0 by Frobenius method with coefficients represented up to and including x3？ Since & 代入ODE   **取最小冪次項 xr1 ( i.e. m=0)之係數為零 indicial eq. r(r1)r=0  r2=0  r = 0 (case 2.)

  13. 5.4 Power FrobeniusMethod(11/20) (when r=0) 級數式 註標變換 (第2,4項 m1=s, m=s+1；其它項 m=s)  (s+1)2 as+1 + (s+1)2as=0  recurrence as+1 = as a0= a1 =a2= a3 =a4= … 取a0=1 

  14. 5.4 Power FrobeniusMethod(12/20) (second solution) By the method of reduction of order (x2x)y”+(3x1)y’ +y= 0 y”+(3x1)/(x2x)y’ + 1/(x2x)y= 0 y2 = uy1 where u =  Udx &  &   ＃

  15. 5.4 Power FrobeniusMethod(13/20) *(second solution) By the method of undetermined coefficients  & 代回原式 &  整理得 其中 註標變換

  16. 5.4 Power FrobeniusMethod(14/20) S=0 –A1 =0  A1 =0 s=1  (-2A2+3A1-2A2+A1)=0  (-4A2+4A1)=0  A2 =0 s≧2   As+1=As  A1 =A2 =A3 =….= 0 &  ＃

  17. THEOREM 2 Frobenius Method. Basis of Solutions. Three Cases(3) Case 3. Roots Differing by an Integer. A basis is (9) (of the same general form as before) and (10) where the roots are so denoted that r1– r2 > 0 and k may turn out to be zero. 5.4 Power FrobeniusMethod(15/20)

  18. 5.4 Power FrobeniusMethod(16/20) • Example (C) : Solve (x2x)y”xy’ + y = 0 by Frobenius method with coefficients represented up to and including x3？ Since & 代入ODE   **取最小冪次項 xr1 ( i.e. m=0)之係數為零 indicial eq.  r(r1)=0  r = 1 or r = 0 (case 3.)

  19. 5.4 Power FrobeniusMethod(17/20) (when r1=1) 級數式 註標變換 (第1,3,4項 m+1=s, m=s 1；其它項 m=s)   [s(s  1)  s +1] as-1(s+1)sas = 0  recurrence a1 = 0‧a0 = 0 a1 = 0 & also a2 = a3 =a4 = … = 0 取a0=1  y1 = x1a0 = x

  20. 5.4 Power FrobeniusMethod(18/20) (second solution) By the method of reduction of order (x2x)y” xy’ +y= 0 y” x/(x2x)y’ + 1/(x2x)y= 0 y2 = uy1 where u =  Udx &  &   ＃

  21. 5.4 Power FrobeniusMethod(19/20) *(second solution) r2=0 By the method of undetermined coefficients & & 代回原式  &  整理得 其中 註標變換

  22. 5.4 Power FrobeniusMethod(20/20) s=0常數-k + A0 = 0  A0 = k s=1  –2A2–A1 + A1 =0  A2 =0 s≧2   A3 = A2/6 = 0, A4 = A3/3 = 0 ...  A2 =A3 =A4 =….= 0 & A0 = k , A1 = unknown  取 A0=1及 去除y1=x 部分仍為基底  ＃

  23. 5.5 Bessel’s Equation. Bessel Functions Jν(1/11) One of the most important ODEs in applied mathematics is Bessel’s equation, (1)x2y" + xy' + (x2 –ν2)y = 0 The parameter ν in (1) is a given number. We assume that ν is real and nonnegative. ＊The solutions ofBessel’s equationareBessel’s functions. ＊By Frobenius method we can derive the indical equation as(r +ν)(r –ν) = 0& The roots arer1 = ν(≧ 0) andr2 = –ν

  24. 5.5 Bessel’s Equation. Bessel Functions Jν(2/11) One of the most important ODEs in applied mathematics is Bessel’s equation, (1)x2y" + xy' + (x2 –ν2)y = 0 The parameter ν in (1) is a given number. We assume that ν is real and nonnegative. ＊The solutions of Bessel’s equation are Bessel’s functions.

  25. 5.5 Bessel’s Equation. Bessel Functions Jν(3/11) If the integer values of ν are denoted by n, For ν=n the relation (7) becomes (8) With particular and n!(n + 1) ‥‥ (n +m) = (m +n)! so that (8) simply becomes (10)

  26. 5.5 Bessel’s Equation. Bessel Functions Jν(4/11) Then a particular solution of (1), denoted by Jn(x) and given by (11) Jn(x)is called the Bessel function of the first kind of order n.（n階第一類貝索函數）The series (11) converges for all x .

  27. 5.5 Bessel’s Equation. Bessel Functions Jν(5/11) For n = 0 we obtain the Bessel function of order 0 (12) which looks similar to a cosine. For n = 1 we obtain the Bessel function of order 1 (13) .

  28. 5.5 Bessel’s Equation. Bessel Functions Jν(6/11) Bessel FunctionsJν(x)for any ν≥ 0. Then (19) With these coefficients and r =r1 =νwe get a particular solution, denoted by Jν(x) and given by (20) Jν(x) is called the Bessel function of the first kind of order ν. The series converges for all x

  29. THEOREM 1 General Solution of Bessel’s Equation THEOREM 2 Linear Dependence of Bessel Functions Jn and J–n If ν is not an integer, a general solution of Bessel’s equation for all x ≠ 0 is (22) y(x) = c1Jν(x) + c2J –ν(x) For integer ν = n the Bessel functions Jn(x) and J–n(x) are linearly dependent, because (23) J–n(x) = (–1)nJn(x) (n = 1, 2, ‥‥). 5.5 Bessel’s Equation. Bessel Functions Jν(7/11)

  30. THEOREM 3 Derivatives, Recursions The derivative of Jν(x) with respect to x can be expressed by Jν-1(x) or Jν+1(x) by the formulas (24) Furthermore, Jν(x) and its derivative satisfy the recurrence relations (24) 5.5 Bessel’s Equation. Bessel Functions Jν(8/11)

  31. THEOREM 4 Elementary Jν for Half-Integer Order ν Bessel functions Jν of orders ±1/2, ±3/2, ±5/2 ‥‥are elementary; they can be expressed by finitely many cosines and sines and powers of x. In particular, (25) 5.5 Bessel’s Equation. Bessel Functions Jν(9/11)

  32. 5.5 Bessel’s Equation. Bessel Functions Jν(10/11) From (24c) with ν= 1/2 and ν= –1/2 and (25) we obtain respectively, and so on.

  33. 5.5 Bessel’s Equation. Bessel Functions Jν(11/11) 利用參數變換使非標準 Bessel’s equation轉換成標準形式 Example (A) : x2y" + xy' + (2x2ν2)y = 0 令 x = z  x = z/ & dz/dx =  So: & 代回原式   ＃ Where if  is not an integer  Y (z) = c1J(z) + c2 J(z) Or y(x) = c1J(x) + c2 J(x)＃

  34. 5.6 Bessel Functions of the Second Kind Yν(x)(1/7) • n = 0: Bessel Function of the Second Kind Y0(x) When n = 0, Bessel’s equation can be written (1) xy" + y' + xy = 0 Then the indicial equation has a double root r = 0. This is Case 2. In this case we first have only one solution, J0(x). We see that the desired second solution must be of the form

  35. 5.6 Bessel Functions of the Second Kind Yν(x)(2/7) • n = 0: Bessel Function of the Second Kind Y0(x) By comparing the coefficient, we find A1 = A3 = ‥‥ = 0 Then, we obtain the result

  36. 5.6 Bessel Functions of the Second Kind Yν(x)(3/7) • Since J0 andy2 are linearly independent functions, they form a basis of (1) for x > 0. • Of course, another basis is obtained if we replace y2 by an independent particular solution of the form a(y2 + bJ0), where a (≠ 0) and b are constants. It is customary to choose a = 2π and b =γ– ln 2, where the number γ = 0.577 215 664 90 ‥‥ is the so-called Euler constant. • The standard particular solution thus obtained is called the Bessel function of the second kind of order zero is denoted by Y0(x)

  37. 5.6 Bessel Functions of the Second Kind Yν(x)(4/7) • For this reason we introduce a standard second solution Y (x) defined for all ν by the formula which is called the Bessel function of the second kind of order ν denoted by Yν(x).

  38. THEOREM 1 General Solution of Bessel’s Equation A general solution of Bessel’s equation for all values of ν (and x > 0) is (9)y(x) = C1Jν(x) + C2Yν(x) 5.6 Bessel Functions of the Second Kind Yν(x)(5/7)

  39. 5.6 Bessel Functions of the Second Kind Yν(x)(6/7) 利用參數變換使非標準 Bessel’s equation轉換成標準形式 Example (B) : xy"  y' + 4xy = 0 令xu = y & 2x=z x = z/2 & dz/dx = 2 So: & Also, & 代回原式    ＃ Where =1 U(z) = c1J1(z) + c2Y1(z) Or y(x) = x [c1J1(2x) + c2Y1(2x)]＃

  40. 5.6 Bessel Functions of the Second Kind Yν(x)(7/7) For the modifiedBessel’s equation, x2y" + xy'–(x2 + ν2)y = 0 ＊The solutions of modifiedBessel’s equation are modifiedBessel’s functions. i.e. y(x) = C1Iν(x) + C2Kν(x) Where &

More Related