High dimensional nonlinear diffusion stochastic processes: modelling for engineering applications
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2001
|
| Series: | Series on advances in mathematics for applied sciences
56 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 297 S. |
| ISBN: | 9810243855 |
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| adam_text | SERIES ON ADVANCES IN MATHEMATICS FOR APPLIED SCIENCES - VOL. 56
HIGH-DIMENSIONAL NONLINEAR DIFFUSION STOCHASTIC PROCESSES MODELLING FOR
ENGINEERING APPLICATIONS YEVGENY MAMONTOV MAGNUS WILLANDER DEPARTMENT OF
PHYSICS, MC2 GOTHENBURG UNIVERSITY AND CHALMERS UNIVERSITY OF TECHNOLOGY
SWEDEN WORLD SCIENTIFIC SINGAPORE * NEW JERSEY * LONDON * HONG KONG
CONTENTS PREFACE VII CHAPTER 1 INTRODUCTORY CHAPTER 1 1.1 PREREQUISITES
FOR READING 1 1.2 RANDOM VARIABLE. STOCHASTIC PROCESS. RANDOM FIELD.
HIGH-DIMENSIONAL PROCESS. ONE-POINT PROCESS 3 1.3 TWO-POINT PROCESS.
EXPECTATION. MARKOV PROCESS. EXAMPLE OF NON-MARKOV PROCESS ASSOCIATED
WITH MULTIDIMENSIONAL MARKOV PROCESS 10 1.4 PRECEDING, SUBSEQUENT AND
TRANSITION PROBABILITY DENSITIES. THE CHAPMAN*KOLMOGOROV EQUATION.
INITIAL CONDITION FOR MARKOV PROCESS 16 1.4.1 THE CHAPMAN-KOLMOGOROV
EQUATION 19 1.4.2 INITIAL CONDITION FOR MARKOV PROCESS 20 1.5
HOMOGENEOUS MARKOV PROCESS. EXAMPLE OF MARKOV PROCESS: THE WIENER
PROCESS 23 1.6 EXPECTATION, VARIANCE AND STANDARD DEVIATIONS OF MARKOV
PROCESS 26 1.7 INVARIANT AND STATIONARY MARKOV PROCESSES. COVARIANCE.
SPECTRAL DENSITIES 30 1.8 DIFFUSION PROCESS 37 XIV CONTENTS 1.9 EXAMPLE
OF DIFFUSION PROCESSES: SOLUTIONS OF ITO S STOCHASTIC ORDINARY
DIFFERENTIAL EQUATION 40 1.10 THE KOLMOGOROV BACKWARD EQUATION 46 1.11
FIGURES OF MERIT. DIFFUSION MODELLING OF HIGH-DIMENSIONAL SYSTEMS 48
1.12 COMMON ANALYTICAL TECHNIQUES TO DETERMINE PROBABILITY DENSITIES OF
DIFFUSION PROCESSES. THE KOLMOGOROV FORWARD EQUATION 51 1.12.1
PROBABILITY DENSITY 51 1.12.2 INVARIANT PROBABILITY DENSITY 54 1.12.3
STATIONARY PROBABILITY DENSITY 57 1.13 THE PURPOSE AND CONTENT OF THIS
BOOK 60 CHAPTER 2 DIFFUSION PROCESSES 63 2.1 INTRODUCTION 63 2.2
TIME-DERIVATIVES OF EXPECTATION AND VARIANCE 64 2.3 ORDINARY
DIFFERENTIAL EQUATION SYSTEMS FOR EXPECTATION . . 66 2.3.1 THE
FIRST-ORDER SYSTEM 66 2.3.2 THE SECOND-ORDER SYSTEM 68 2.3.3 SYSTEMS OF
THE HIGHER ORDERS 70 2.4 MODELS FOR NOISE-INDUCED PHENOMENA IN
EXPECTATION 71 2.4.1 THE CASE OF STOCHASTIC RESONANCE 71 2.4.2
PRACTICALLY EFFICIENT IMPLEMENTATION OF THE SECOND-ORDER SYSTEM 73 2.5
ORDINARY DIFFERENTIAL EQUATION SYSTEM FOR VARIANCE 76 2.5.1 DAMPING
MATRIX 76 2.5.2 THE UNCORRELATED-MATRIXES APPROXIMATION 77 _.. 2.5.3
NONLINEARITY OF THE DRIFT FUNCTION 80 2.5.4 FUNDAMENTAL LIMITATION OF
THE STATE-SPACE- INDEPENDENT APPROXIMATIONS FOR THE DIFFUSION AND
DAMPING MATRIXES 81 2.6 THE STEADY-STATE APPROXIMATION FOR THE
PROBABILITY DENSITY 82 CHAPTER 3 INVARIANT DIFFUSION PROCESSES 85 3.1
INTRODUCTION 85 3.2 PRELIMINARY REMARKS 85 3.3 EXPECTATION. THE
FINITE-EQUATION METHOD 86 CONTENTS XV 3.4 EXPLICIT EXPRESSION FOR
VARIANCE 88 3.5 THE SIMPLIFIED DETAILED-BALANCE APPROXIMATION FOR
INVARIANT PROBABILITY DENSITY 90 3.5.1 PARTIAL DIFFERENTIAL EQUATION FOR
LOGARITHM OF THE DENSITY 90 3.5.2 TRUNCATED EQUATION FOR THE LOGARITHM
AND THE DETAILED-BALANCE EQUATION 91 3.5.3 CASE OF THE DETAILED BALANCE
93 3.5.4 THE DETAILED-BALANCE APPROXIMATION 95 3.5.5 THE SIMPLIFIED
DETAILED-BALANCE APPROXIMATION. THEOREM ON THE APPROXIMATING DENSITY 96
3.6 ANALYTICAL-NUMERICAL APPROACH TO NON-INVARIANT AND INVARIANT
DIFFUSION PROCESSES 99 3.6.1 CHOICE OF THE BOUNDED DOMAIN OF THE
INTEGRATION .. 100 3.6.2 EVALUATION OF THE MULTIFOLD INTEGRALS. THE
MONTE CARLO TECHNIQUE 102 3.6.3 SUMMARY OF THE APPROACH 104 3.7
DISCUSSION 105 CHAPTER 4 STATIONARY DIFFUSION PROCESSES 107 4.1
INTRODUCTION .*- 107 4.2 PREVIOUS RESULTS RELATED TO COVARIANCE AND
SPECTRAL-DENSITY MATRIXES 108 4.3 TIME-SEPARATION DERIVATIVE OF
COVARIANCE IN THE LIMIT CASE OF ZERO TIME SEPARATION 109 4.4 FLICKER
EFFECT ILL 4.5 TIME-SEPARATION DERIVATIVE OF COVARIANCE IN THE GENERAL
CASE 112 4.6 CASE OF THE UNCORRELATED MATRIXES 114 4.7 REPRESENTATIONS
FOR SPECTRAL DENSITY IN THE UNCORRELATED-MATRIXES CASE 117 4.8 EXAMPLE:
COMPARISON OF THE DAMPINGS FOR A PARTICLE NEAR THE MINIMUM OF ITS
POTENTIAL ENERGY 118 4.9 THE DETERMINISTIC-TRANSITION APPROXIMATION 123
4.10 EXAMPLE: NON-EXPONENTIAL COVARIANCE OF VELOCITY OF A PARTICLE IN A
FLUID 126 4.10.1 COVARIANCE IN THE GENERAL CASE 126 XVI CONTENTS 4.10.2
COVARIANCE AND THE QUATITIES RELATED TO IT IN A SIMPLE FLUID 128 4.10.3
CASE OF THE HARD-SPHERE FLUID 130 4.10.4 SUMMARY OF THE PROCEDURE IN THE
GENERAL CASE .. 133 4.11 ANALYTICAL-NUMERICAL APPROACH TO STATIONARY
PROCESS .... 134 4.11.1 PRACTICAL ISSUES 134 4.11.2 SUMMARY OF THE
APPROACH 136 4.12 DISCUSSION 137 CHAPTER 5 ITO S STOCHASTIC PARTIAL
DIFFERENTIAL 141 EQUATIONS AS NON-MARKOV MODELS LEADING TO
HIGH-DIMENSIONAL DIFFUSION PROCESSES 5.1 INTRODUCTION 141 5.2 VARIOUS
TYPES OF ITO S STOCHASTIC DIFFERENTIAL EQUATIONS . . . 142 5.3 METHOD TO
REDUCE ISPIDE TO SYSTEM OF ISODES 144 5.3.1 PROJECTION APPROACH 148
5.3.2 STOCHASTIC COLLOCATION METHOD 151 5.3.3
STOCHASTIC-ADAPTIVE-INTERPOLATION METHOD 153 5.4 RELATED COMPUTATIONAL
ISSUES 159 5.5 DISCUSSION 160 CHAPTER 6 ITO S STOCHASTIC PARTIAL
DIFFERENTIAL 163 EQUATIONS FOR ELECTRON FLUIDS IN SEMICONDUCTORS 6.1
INTRODUCTION 163 6.2 MICROSCOPIC PHENOMENA IN MACROSCOPIC MODELS OF
MULTIPARTICLE SYSTEMS 165 6.2.1 MICROSCOPIC RANDOM WALKS IN
DETERMINISTIC MACROSCOPIC MODELS OF MULTIPARTICLE SYSTEMS 165 6.2.2
MACROSCALE, MESOSCALE AND MICROSCALE DOMAINS IN TERMS OF THE
WAVE-DIFFUSION EQUATION 171 6.2.3 STOCHASTIC GENERALIZATION OF THE
DETERMINISTIC MACROSCOPIC MODELS OF MULTIPARTICLE SYSTEMS 176 6.3 THE
ISPDE SYSTEM FOR ELECTRON FLUID IN N-TYPE SEMICONDUCTOR 177 6.3.1
DETERMINISTIC MODEL FOR ELECTRON FLUID IN SEMICONDUCTOR 177 CONTENTS
XVII 6.3.2 MESOSCOPIC WAVE-DIFFUSION EQUATIONS IN THE DETERMINISTIC
FLUID-DYNAMIC MODEL 180 6.3.3 STOCHASTIC GENERALIZATION OF THE
DETERMINISTIC FLUID-DYNAMIC MODEL. THE SEMICONDUCTOR-FLUID ISPDE SYSTEM
187 6.4 SEMICONDUCTOR NOISES AND THE SF-ISPDE SYSTEM: DISCUSSION AND
FUTURE DEVELOPMENT 192 6.4.1 THE SF-ISPDE SYSTEM IN CONNECTION WITH
SEMICONDUCTOR NOISES 192 6.4.2 SOME DIRECTIONS FOR FUTURE DEVELOPMENT
195 CHAPTER 7 DISTINGUISHING FEATURES 197 OF ENGINEERING APPLICATIONS
7.1 HIGH-DIMENSIONAL DIFFUSION PROCESSES 197 7.2 EFFICIENT MULTIPLE
ANALYSIS 198 7.3 REASONABLE AMOUNT OF THE MAIN COMPUTER MEMORY 198 7.4
REAL-TIME SIGNAL TRANSFORMATION BY DIFFUSION PROCESS 199 CHAPTER 8
ANALYTICAL-NUMERICAL APPROACH 201 TO ENGINEERING PROBLEMS C AND COMMON
ANALYTICAL TECHNIQUES 8.1 ANALYTICAL-NUMERICAL APPROACH TO ENGINEERING
PROBLEMS 201 8.2 SEVERE PRACTICAL LIMITATIONS OF COMMON ANALYTICAL
TECHNIQUES IN THE HIGH-DIMENSIONAL CASE. POSSIBLE ALTERNATIVES 203
APPENDIX A EXAMPLE OF MARKOV PROCESSES: 205 SOLUTIONS OF THE CAUCHY
PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATION SYSTEM APPENDIX B
SIGNAL-TO-NOISE RATIO 209 APPENDIX C EXAMPLE OF APPLICATION OF COROLLARY
1.2: 213 NONLINEAR FRICTION AND UNBOUNDED STATIONARY PROBABILITY DENSITY
OF THE PARTICLE VELOCITY IN UNIFORM FLUID C.I DESCRIPTION OF THE MODEL
213 XVIII CONTENTS C.2 ENERGY-INDEPENDENT MOMENTUM-RELAXATION TIME.
EQUILIBRIUM PROBABILITY DENSITY 220 C.3 ENERGY-DEPENDENT
MOMENTUM-RELAXATION TIME: GENERAL CASE 222 C.4 ENERGY-DEPENDENT
MOMENTUM-RELAXATION TIME: CASE OF SIMPLE FLUID 227 APPENDIX D PROOFS OF
THE THEOREMS IN CHAPTER 2 231 AND OTHER DETAILS D.I PROOF OF THEOREM 2.1
231 D.2 PROOF OF THEOREM 2.2 232 D.3 GREEN S FORMULA FOR THE
DIFFERENTIAL OPERATOR OF KOLMOGOROV S BACKWARD EQUATION 235 D.4 PROOF OF
THEOREM 2.3 237 D.5 QUASI-NEUTRAL EQUILIBRIUM POINT 239 D.6 PROOF OF
THEOREM 2.4 241 APPENDIX E PROOFS OF THE THEOREMS IN CHAPTER 4 243 E.I
PROOF OF LEMMA 4.1 243 E.2 PROOF OF THEOREM 4.2 243 E.3 PROOF OF THEOREM
4.3 244 E.4 PROOF OF THEOREM 4.4 245 APPENDIX F HIDDEN RANDOMNESS IN
NONRANDOM 247 EQUATION FOR THE PARTICLE CONCENTRATION OF UNIFORM FLUID
AND CHEMICAL-REACTION /GENERATION-RECOMBINATION NOISE APPENDIX G
EXAMPLE: EIGENVALUES AND EIGENFUNCTIONS 255 OF THE LINEAR DIFFERENTIAL
OPERATOR ASSOCIATED WITH A BOUNDED DOMAIN IN THREE-DIMENSIONAL SPACE
APPENDIX H RESOURCES FOR ENGINEERING PARALLEL 261 COMPUTING UNDER
WINDOWS 95 BIBLIOGRAPHY 265 INDEX ^ 281
|
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| id | DE-604.BV013736619 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:51:06Z |
| institution | BVB |
| isbn | 9810243855 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009389196 |
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| physical | XVIII, 297 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | World Scientific |
| record_format | marc |
| series | Series on advances in mathematics for applied sciences |
| series2 | Series on advances in mathematics for applied sciences |
| spelling | Mamontov, Yevgeny Verfasser aut High dimensional nonlinear diffusion stochastic processes modelling for engineering applications Yevgeny Mamontov ; Magnus Willander High-dimensional nonlinear diffusion stochastic processes Singapore [u.a.] World Scientific 2001 XVIII, 297 S. txt rdacontent n rdamedia nc rdacarrier Series on advances in mathematics for applied sciences 56 Ingénierie - Modèles mathématiques Processus de diffusion Processus stochastiques Équations différentielles non linéaires Ingenieurwissenschaften Mathematisches Modell Differential equations, Nonlinear Diffusion processes Engineering Mathematical models Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Nichtlineare Diffusionsgleichung (DE-588)4171749-1 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s Nichtlineare Diffusionsgleichung (DE-588)4171749-1 s DE-604 Willander, Magnus Verfasser aut Series on advances in mathematics for applied sciences 56 (DE-604)BV004569239 56 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009389196&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mamontov, Yevgeny Willander, Magnus High dimensional nonlinear diffusion stochastic processes modelling for engineering applications Series on advances in mathematics for applied sciences Ingénierie - Modèles mathématiques Processus de diffusion Processus stochastiques Équations différentielles non linéaires Ingenieurwissenschaften Mathematisches Modell Differential equations, Nonlinear Diffusion processes Engineering Mathematical models Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Nichtlineare Diffusionsgleichung (DE-588)4171749-1 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4171749-1 |
| title | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications |
| title_alt | High-dimensional nonlinear diffusion stochastic processes |
| title_auth | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications |
| title_exact_search | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications |
| title_full | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications Yevgeny Mamontov ; Magnus Willander |
| title_fullStr | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications Yevgeny Mamontov ; Magnus Willander |
| title_full_unstemmed | High dimensional nonlinear diffusion stochastic processes modelling for engineering applications Yevgeny Mamontov ; Magnus Willander |
| title_short | High dimensional nonlinear diffusion stochastic processes |
| title_sort | high dimensional nonlinear diffusion stochastic processes modelling for engineering applications |
| title_sub | modelling for engineering applications |
| topic | Ingénierie - Modèles mathématiques Processus de diffusion Processus stochastiques Équations différentielles non linéaires Ingenieurwissenschaften Mathematisches Modell Differential equations, Nonlinear Diffusion processes Engineering Mathematical models Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Nichtlineare Diffusionsgleichung (DE-588)4171749-1 gnd |
| topic_facet | Ingénierie - Modèles mathématiques Processus de diffusion Processus stochastiques Équations différentielles non linéaires Ingenieurwissenschaften Mathematisches Modell Differential equations, Nonlinear Diffusion processes Engineering Mathematical models Stochastic processes Stochastischer Prozess Nichtlineare Diffusionsgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009389196&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004569239 |
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