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CHAPTER 4. THEORY 89 So, we have Kmax |G2(u)(x, k) − G2(u)(y, k)|dk 0 e −Cx x C Cz k e k dz + y k C −Cy [e k − e k −Cx y k ] e 0 Cz k dz dk = 4τ Kmax 2KmaxT ∞uX 1 − e h 0 C(y−x) k + e −Cy k − e −Cx k e Cy k − 1 dk = 4τ Kmax 2KmaxT ∞uX 1 − e h 0 C(y−x) k + 1 − e C(y−x) k − e −Cy k + e −Cx k dk = 4τ Kmax 2KmaxT ∞uX 2 − 2e h C(y−x) k + e −Cx k − e −Cy k dk. ≤ 4τ Kmax 2KmaxT ∞uX h 0 We will now break the k interval (0,Kmax] into two intervals. Choose ɛk so that Kmax > ɛk > 0. Then the two intervals are (0,ɛk) and[ɛk,Kmax]. For the first interval we have: 4τ ɛk 2KmaxT ∞uX 2 − 2e h 0 C(y−x) k + e −Cx k − e −Cy k dk ≤ 4τ ɛk 2KmaxT ∞uX |2 − 2e h 0 C(y−x) k + e −Cx k − e −Cy k |dk ≤ 4τ ɛk 2KmaxT ∞uX 2+2e h 0 C(y−x) k + e −Cx k + e −Cy k dk ≤ 4τ ɛk 2KmaxT ∞uX [2+2+1+1]dk h 0 = 24τ 2KmaxT ∞uXɛk. h 0
CHAPTER 4. THEORY 90 We want to choose ɛk so that the bound over this first interval is equal to γ 4 .Onthe second interval we have: 4τ Kmax 2KmaxT ∞uX h ɛk ≤ 4τ Kmax 2KmaxT ∞uX h ɛk ≤ 12τ Kmax 2KmaxT ∞uX h = 12τ h 2 − 2e C(y−x) k + e −Cx k − e −Cy k dk |2 − 2e C(y−x) ɛ k | + e −Cy k |e C(y−x) ɛ k − 1|dk |1 − e C(y−x) ɛ k |dk ɛk 2KmaxT ∞uX(Kmax − ɛk)|1 − e C(y−x) ɛk | ≤ 12τ h Kmax 2KmaxT ∞uX|1 − e C(y−x) ɛk |. We want to choose x and y close enough together so that this bound over the second interval is equal to γ Cx ɛ . We can do this since e k is a continuous function. Therefore, 4 we have shown that we can choose x and y close enough so that Kmax 0 |G2(u)(x, k) 2 − G2(u)(y, k) 2 |dk ≤ γ 2 . Using a very similar argument for the case 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 51 and 52: CHAPTER 2. TEMPORAL INTEGRATION 38
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: CHAPTER 4. THEORY 88 So an estimate
Page 105 and 106: CHAPTER 4. THEORY 92 The exact same
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
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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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