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CHAPTER 4. THEORY 91 Thisshowsthatifwechoosex and y closetogether,wehavethat Kmax |G2(u)(x, k) 2 − G2(u)(y, k) 2 |dk ≤ γ. −Kmax So Kmax −Kmax G2(u)(x, k) 2 is a continuous function. This means that G1(u),G2(u) ∈ X for u ∈ X, and therefore, Z maps X to X. Show Z is continuous Let {un} ⊂X with un → u ∈ X. To show that Z is continuous, we need to show Z(un) → Z(u), or G1(un) → G1(u) andG2(un) → G2(u). Now for the G1 term we have for k>0, |G1(u)(x, k) − G1(un)(x, k)| = C k x 0 dze C(z−x) k h(z,k) ¯ Kmax −Kmax x ≤ dz 0 C C(z−x) e k | k ¯ Kmax h(z,k)| −Kmax x ≤ dz 0 C C(z−x) e k k ¯ h∞ 2KmaxuX = ¯ Cx − h∞ 2Kmaxu − unX 1 − e k ≤ ¯ h∞ 2Kmaxu − unX, [u(z,k ′ ) − un(z,k ′ )]dk ′ |u(z,k ′ ) − un(z,k ′ )|dk ′
CHAPTER 4. THEORY 92 The exact same bound holds for k0that G2(un)(x, k) − G2(u)(x, k) = 4Cτ hk − 4Cτ hk x Kmax 0 −Kmax x Kmax 0 −Kmax This is the same as looking at two separate terms, where G2(un)(x, k) − G2(u)(x, k) = 4Cτ x hk 0 + 4Cτ x hk 0 e C(z−x) k e C(z−x) k Kmax −Kmax Kmax −Kmax e C(z−x) k u(z,k ′ )Tu(z,k − k ′ )dk ′ dz e C(z−x) k un(z,k ′ )Tun(z,k − k ′ )dk ′ dz. Tu(z,k − k ′ )[u(z,k ′ ) − un(x, k)] dk ′ dz [Tu(z,k − k ′ ) − Tun(z,k − k ′ )] dk ′ dzun(z,k ′ ) = F 1 (x, k)+F 2 (x, k).
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Numerical Methods for the Wigner-Po
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NUMERICAL METHODS FOR THE WIGNER-PO
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Acknowledgments First and foremost,
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Table of Contents List of Tables ix
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4.3 ApplicationofSchauder’sFixedP
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List of Tables 1.1 PhysicalConstant
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3.11 I-V Curve with 30 Percent Redu
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CHAPTER 1. OVERVIEW 2 being heavily
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CHAPTER 1. OVERVIEW 4 Doping Spacer
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CHAPTER 1. OVERVIEW 6 and identfyin
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CHAPTER 1. OVERVIEW 8 for predictin
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CHAPTER 1. OVERVIEW 10 by Schrödin
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CHAPTER 1. OVERVIEW 12 ranging term
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CHAPTER 1. OVERVIEW 14 But because
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CHAPTER 1. OVERVIEW 16 = h 2π ,we
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CHAPTER 1. OVERVIEW 18 Table 1.1: P
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CHAPTER 1. OVERVIEW 20 where q is t
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CHAPTER 1. OVERVIEW 22 get K(f)
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CHAPTER 1. OVERVIEW 24 rule. The pr
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CHAPTER 1. OVERVIEW 26 The weights
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Chapter 2 Temporal Integration 2.1
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CHAPTER 2. TEMPORAL INTEGRATION 30
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Page 53 and 54: CHAPTER 2. TEMPORAL INTEGRATION 40
Page 55 and 56: CHAPTER 2. TEMPORAL INTEGRATION 42
Page 57 and 58: Chapter 3 Bifurcation Analysis 3.1
Page 59 and 60: CHAPTER 3. BIFURCATION ANALYSIS 46
Page 87 and 88: Chapter 4 Theory 4.1 Steady-State T
Page 89 and 90: CHAPTER 4. THEORY 76 If we write ou
Page 91 and 92: CHAPTER 4. THEORY 78 So if we write
Page 93 and 94: CHAPTER 4. THEORY 80 Combining Equa
Page 95 and 96: CHAPTER 4. THEORY 82 We now estimat
Page 97 and 98: CHAPTER 4. THEORY 84 pendent of x,
Page 99 and 100: CHAPTER 4. THEORY 86 interval we ha
Page 101 and 102: CHAPTER 4. THEORY 88 So an estimate
Page 103: CHAPTER 4. THEORY 90 We want to cho
Page 107 and 108: CHAPTER 4. THEORY 94 Since un → u
Page 109 and 110: CHAPTER 4. THEORY 96 thereexistsaco
Page 111 and 112: CHAPTER 4. THEORY 98 For 0
Page 113 and 114: CHAPTER 4. THEORY 100 denote this d
Page 115 and 116: CHAPTER 4. THEORY 102 Now, we can t
Page 117 and 118: CHAPTER 4. THEORY 104 operator. 4.2
Page 119 and 120: CHAPTER 4. THEORY 106 we have at th
Page 121 and 122: CHAPTER 4. THEORY 108 we have at th
Page 123 and 124: CHAPTER 4. THEORY 110 4.3 Applicati
Page 125 and 126: CHAPTER 4. THEORY 112 We need to fi
Page 127 and 128: Chapter 5 Conclusion In this work,
Page 129 and 130: CHAPTER 5. CONCLUSION 116 • Findi
Page 131 and 132: REFERENCES 118 ODE solver. Technica
Page 133 and 134: REFERENCES 120 [21] C. T. Kelley an
Page 135 and 136: REFERENCES 122 [37] Homer F. Walker
Page 137 and 138: APPENDIX A. NOTATION 124 ˜B Genera
Page 139 and 140: APPENDIX A. NOTATION 126 k Wave num
Page 141 and 142: APPENDIX A. NOTATION 128 ∆t Incre
Page 143 and 144: APPENDIX A. NOTATION 130 ɛc ɛk ɛ
Page 145 and 146: APPENDIX B. TRILINOS 132 user-given
Page 147 and 148: APPENDIX B. TRILINOS 134 Trilinos,
Page 149 and 150: APPENDIX B. TRILINOS 136 Minimum Re
Page 151 and 152: APPENDIX B. TRILINOS 138 A very use
Page 153 and 154: Appendix C Guide to RTD Simulation
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APPENDIX C. GUIDE TO RTD SIMULATION
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Appendix D GMRES and Arnoldi Iterat
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APPENDIX D. GMRES AND ARNOLDI ITERA
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APPENDIX D. GMRES AND ARNOLDI ITERA
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