Solving Differential Equations in Terms of Bessel Functions

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50 CHAPTER 3. SOLVING IN TERMS OF BESSEL FUNCTIONS Example 3.2 As in Example 2.11 we start with the modified Bessel operator LB with ν = 2. We do the same change of variables with f = 2(x − 1)(x − 2)2 (x − 3) 2 and get the operator > LB:=xˆ2*Dxˆ2+x*Dx-(xˆ2+2ˆ2): > f:=2*(x-1)*(x-2)ˆ2/(x-3)ˆ2: > L:=changeOfVars(LB,f); = 2x3 − 10x 2 + 16x − 8 x 2 − 6x + 9 L :=(x − 2) 3 x 2 − 7x + 8 (x − 3) 6 (x − 1) 3 ∂ 2 + 4 3 2 5 2 2 x − 14x + 55x − 84x + 46 (x − 3) (x − 1) (x − 2) ∂− 4 x 2 3 − 7x + 8 x 6 − 10x 5 + 42x 4 − 100x 3 + 158x 2 − 172x + 97 (x − 2)(x − 1) The singularities of this operator are 1,2,3,∞, 7 2 + 1 √ 7 2 17 and 2 − 1 √ 2 17. We already discovered in Example 2.11 that the latter two singularities are apparent, 1 and 2 are the regular singularities, and 3 and ∞ are irregular singularities. We will now use part (b) of the theorem to find f . In order to do this we compute the generalized exponent at x = 3 using Maple: > gen exp(L,t,x=3); [[− 8 10 8 10 − + 1,t = x − 3],[ + + 1,t = x − 3]] (3.9) t2 t t2 t The exponent difference is ∆(L,3) = − 16 t 2 3 − 20 . t3 If we divide ∆(L,3) by two and each coefficient by its corresponding degree we get f3 = − 4 t 2 3 − 10 . t3 This will be the polar part corresponding to t3 in the partial fraction decomposition of f . We do the same computations at the point x = ∞: > gen exp(L,t,x=infinity); [[− 2 t 1 1 + ,t = 2 x ],[2 t + 1 2 1 ,t = ]] (3.10) x
3.1. CHANGE OF VARIABLES 51 ⇒ ∆(L,∞) = − 4t −1 ∞ . Hence, f∞ = −2t −1 ∞ = −2x is the polar part corresponding to t∞. Maple’s output in (3.9) and (3.10) is not ordered. Therefore, we defined ∆ modulo a factor −1. So, we don’t know whether the coefficients of f3 and f∞ are 1 or −1 and we also don’t know the constant part of f . But from the polar parts f3 and f∞ we get candidates for the parameter of the change of variables modulo some constant. The solution must be among those candidates. We also know that the regular singularities 1 and 2 must be zeros of f . So we compute: f3 + f∞| x=1 = 6, f3 + f∞| x=2 = 10, f3 − f∞| x=1 = 2, f3 − f∞| x=2 = 2. We do not need the candidates − f3 + f∞ and − f3 − f∞ since LB −x −→C LB. In other words, if we find a solution where ˜f is the parameter of the change of variables, we will also find a solution if we take the parameter − ˜f in the change of variables. In the first candidate f3 + f∞ we would need two different constants in order to make both x = 1 and x = 2 a zero of f . So the only candidate that remains is ˜f = f3 − f∞ − 2 = − 4 10 − (x − 3) 2 x − 3 − 2x − 2 = −2 x3 − 5x2 + 8x − 4 (x − 3) 2 which, in fact, is equal to − f . As a result we can represent the solutions of L with the modified Bessel functions I2( ˜f ) and K2( ˜f ). At the end of the last chapter we defined exp-apparent points. These are singularities p of an operator L where ∆(L, p) ∈ Z and the solutions at p are not logarithmic. We will now distinguish between two more cases which will correspond to zeros and poles of f . Definition 3.3 Let p be a singularity of the operator L ∈ K[∂] which is not expapparent. Then p is called (i) exp-regular ⇔ ∆(L, p) ∈ C, (ii) exp-irregular ⇔ ∆(L, p) ∈ C[1/tp]\C. We denote the set of singularities that are exp-regular by Sreg and those that are exp-irregular by Sirr.
Page 1 and 2: Solving Differential Equations in T
Page 3 and 4: Contents List of Notations v Introd
Page 5 and 6: List of Notations This is a list of
Page 7 and 8: Introduction Ordinary differential
Page 9 and 10: 1 Preliminaries We will first intro
Page 11 and 12: 1.2. SINGULAR POINTS 11 A linear di
Page 13 and 14: 1.3. HYPERGEOMETRIC SERIES 13 Theor
Page 15 and 16: 1.3. HYPERGEOMETRIC SERIES 15 1.3.1
Page 17 and 18: 1.3. HYPERGEOMETRIC SERIES 17 Remar
Page 19 and 20: 1.4. FORMAL SOLUTIONS AND GENERALIZ
Page 27 and 28: 2 Transformations From now on we wi
Page 29 and 30: 2.1. OPERATORS OF DEGREE TWO 29 in
Page 31 and 32: 2.1. OPERATORS OF DEGREE TWO 31 r 3
Page 33 and 34: 2.1. OPERATORS OF DEGREE TWO 33 The
Page 35 and 36: 2.1. OPERATORS OF DEGREE TWO 35 The
Page 37 and 38: 2.2. THE EXPONENT DIFFERENCE 37 > g
Page 39 and 40: 2.2. THE EXPONENT DIFFERENCE 39 whe
Page 41 and 42: 2.3. EQUIVALENCE OF DIFFERENTIAL OP
Page 47 and 48: 3 Solving in Terms of Bessel Functi
Page 49: 3.1. CHANGE OF VARIABLES 49 We star
Page 53 and 54: 3.2. FINDING THE PARAMETER ν 53 Ex
Page 55 and 56: 3.2. FINDING THE PARAMETER ν 55 If
Page 57 and 58: 3.3. THE ALGORITHM 57 3.3 The Algor
Page 59 and 60: 3.3. THE ALGORITHM 59 Example 3.12
Page 61 and 62: 3.3. THE ALGORITHM 61 > gen exp(L,t
Page 63 and 64: 3.3. THE ALGORITHM 63 We can also f
Page 65 and 66: 3.3. THE ALGORITHM 65 > equiv(L,M);
Page 67 and 68: 3.3. THE ALGORITHM 67 must be a mul
Page 69 and 70: 3.3. THE ALGORITHM 69 2. Let ai,bi
Page 71 and 72: 3.3. THE ALGORITHM 71 The exp-regul
Page 73 and 74: 3.4. SOLVING OVER A GENERAL FIELD K
Page 81 and 82: 3.5. WHITTAKER FUNCTIONS 81 At x =
Page 83 and 84: 3.6. TWO FINAL EXAMPLES 83 > LW:=D
Page 85 and 86: 3.6. TWO FINAL EXAMPLES 85 Since Wh
Page 87 and 88: 4 Conclusion We developed an algori
Page 89 and 90: A Appendix A.1 Transformations From
Page 91 and 92: A.3. PACKAGE DESCRIPTION 91 coeffic
Page 93 and 94: A.3. PACKAGE DESCRIPTION 93 findWhi
Page 95 and 96: Bibliography [1] ABRAMOV, S. A., BA
Page 97 and 98: BIBLIOGRAPHY 97 [23] VAN HOEIJ, M.
Page 99 and 100: Index adjoint operator, 42, 44 alge

Solving Differential Equations in Terms of Bessel Functions

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