Advanced methods of mathematical physics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Alpha Science Internat.
2008
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 519 S. graph. Darst. |
| ISBN: | 9781842654095 |
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| 245 | 1 | 0 | |a Advanced methods of mathematical physics |c R. S. Kaushal ; D. Parashar |
| 246 | 1 | 3 | |a Mathematical physics |
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Record in the Search Index
| _version_ | 1804137733563613184 |
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| adam_text | ADVANCED METHODS OF MATH MATICAL PHYSICS SECOND EDITION ADVANCED METHODS
OF MATHEMATICAL PHYSICS SECOND EDITION R.S. KAUSHAL * D. PARASHAR ALPHA
SCIENCE INTERNATIONAL LTD. OXFORD, U.K. CONTENTS PREFACE TO THE SECOND
EDITION VII PREFACE TO THE FIRST EDITION IX ABBREVIATIONS, NOTATIONS AND
SYMBOLS XIII 1. GENERAL INTRODUCTION 1 2. THEORY OF FINITE GROUPS 5 2. 1
A BRIEF REVIEW OF SET THEORY 6 CARTESIAN PRODUCTS 6 MAPPINGS 7 BINARY
COMPOSITIONS 7 COMPOSITION TABLES 8 2.2 ABSTRACT GROUPS 8 PRELIMINARIES
8 CYCLIC GROUPS 10 PERMUTATION GROUP (S,,) 12 GROUPS OF SYMMETRY 14
CONJUGATE ELEMENTS AND CLASSES 18 SUBGROUPS 19 CENTER OF A GROUP 20
COSETS 20 NORMAL SUBGROUPS 22 FACTOR GROUPS 23 2.3 HOMOMORPHISMS 24
AUTOMORPHISMS 26 INNER AND OUTER AUTOMORPHISMS 27 GROUP OF AUTOMORPHISMS
27 DIRECT PRODUCT OF GROUPS 28 SEMIDIRECT PRODUCT OF GROUPS 29 2.4 GROUP
REPRESENTATIONS 30 INVARIANT SUBSPACES 32 REDUCIBLE REPRESENTATIONS 32
IRREDUCIBLE REPRESENTATIONS 34 UNITARY REPRESENTATIONS 36 SCHUR S LEMMAS
37 THE ORTHOGONALITY THEOREM 39 CHARACTERS OF A REPRESENTATION 41 2.5
INTRODUCTION TO CONTINOUS GROUPS 42 LIE GROUPS 43 LIE GROUPS OF
TRANSFORMATIONS 44 INFINITE CONTINOUS GROUPS 45 GENERATORS OF A LIE
GROUP 45 XVI CONTENTS- 2.6 APPLICATIONS TO PHYSICAL PROBLEMS 48
PERMUTATION GROUP (5,,) 48 UNITARY GROUP (SU(;I)) 57 SYMMETRY GROUP OF A
SQUARE 55 2.7 SUMMARY AND FURTHER READING 59 PROBLEMS 60 3. RUDIMENTS OF
TOPOLOGY AND DIFFERENTIAL GEOMETRY 63 3.1 PRELIMINARIES 63 DENUMERABLE
AND COUNTABLE SETS 64 LOWER AND UPPER BOUNDS 66 NEIGHBOURHOODS, OPEN AND
CLOSED SETS 66 CONTINUITY 68 LIMIT POINTS 69 BOLZANO-WEIERSTRASS THEOREM
70 ISOLATED, DENSE AND PERFECT SETS 71 3.2 METRIC SPACES 72 EUCLIDEAN
SPACE 73 HILBERT SPACE 74 DISTANCE BETWEEN SETS 74 OPEN AND CLOSED
SPHERES 75 EQUIVALENCE OF METRIC SPACES 76 3.3 TOPOLOGICAL SPACES 76
DEFINITION 76 UNION AND INTERSECTION OF TOPOLOGIES 78 LIMIT POINTS 79
CLOSURE OF A SET 79 INTERIOR, EXTERIOR AND BOUNDARY OF A SET 80 BASE FOR
A TOPOLOGY 82 HOUSDORFF SPACES 82 RELATIVE TOPOLOGIES 82 3.4 COMPACTNESS
83 SOME DEFINITIONS 83 HEINE-BOREL THEOREM 84 3.5 CONNECTEDNESS 86
SEPARATED SETS 86 CONNECTED SETS 86 CONNECTED SPACES 87 3.6 HOMOTOPY 88
HOMOTOPIC PATHS 88 SIMPLY CONNECTED SPACES 90 THE FUNDAMENTAL GROUP 91
3.7 ESSENTIALS OF DIFFERENTIA] GEOMETRY 94 SOME BASIC CONCEPTS AND
DEFINITIONS 94 DIFFERENTIABLE MANIFOLDS 96 DIFFEOMORPHISM 98 VECTOR
FIELDS 98 DIFFERENTIAL FORMS 101 3.8 SUMMARY AND FURTHER READING 103
PROBLEMS 104 -CONTENTS XVII 4. INTEGRAL EQUATIONS, STURM-LIOUVILLE
THEORY AND GREEN S FUNCTIONS 107 4.1 TERMINOLOGY AND DEFINITIONS 707
FREDHOLM INTEGRAL EQUATIONS 108 VOLTERRA INTEGRAL EQUATIONS 109
DIFFERENTIATION OF A FUNCTION UNDER THE INTEGRAL SIGN 110 RELATION
BETWEEN DIFFERENTIAL AND INTEGRAL EQUATIONS 111 4.2 SOLUTION OF INTEGRAL
EQUATIONS 113 THE LIOUVILLE-NEUMANN SERIES METHOD 113 THE FREDHOLM
METHOD 124 THE HILBERT-SCHMIDT THEORY 134 4.3 STURM-LIOUVILLE THEORY 142
ADJOINT DIFFERENTIAL SYTSTEM 143 THE STURM-LIOUVILLE PROBLEMS:
EIGENVALUES AND EIGENFUNCTIONS 145 4.4 THE GREEN S FUNCTIONS 1.48 ,
DETERMINATION OF G(X, T) 149 CONNECTION WITH INHOMOGENEOUS
STURM-LIOUVILLE EQUATION 152 4.5 APPLICTIONS TO PHYSICS PROBLEMS 153 THE
INFLUENCE FUNCTION 154 * . * THE ABEL S INTEGRAL EQUATION 156 * * *
: * *** * 4.6 SUMMARY AND FURTHER READING 158 PROBLEMS 159 , 5.
STOCHASTIC PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS ,.,,.* , ...
162 5.1 RANDOM VARIABLE AND DISTRIBUTION FUNCTION OF RANDOM VARIABLES
163 , * SOME BASIC DEFINITIONS AND RESULTS 163 MULTIDIMENSIONAL
DISTRIBUTION FUNCTIONS 168 FUNCTIONS OF RANDOM VARIABLES 171 , THE
STIELTJES INTEGRAL 174 5.2 NUMERICAL CHARACTERSTICS OF RANDOM VARIABLES:
MOMENTS OF THE DISTRIBUTION FUNCTION 176 MATHEMATICAL EXPECTATION, 176 :
. . . . . * * VARIANCE 178 CO VARIANCE AND CO VARIANCE MATRIX 182
CHARACTERISTIC FUNCTION OF RANDOM VARIABLES 185 5.3 STOCHASTIC
PROCESSES: MARKOV PROCESS 186 CLASSIFICATION OF STOCHASTIC PROCESSES 187
MARKOV PROCESSES: FOKKER-PLANCK EQUATION 191 GENERAL THEORY OF
CONTINUOUS (MARKOV) PROCESSES 193 , . . 5.4 STOCHASTIC DIFFERENTIAL
EQUATIONS: AN INTRODUCTION 799 5.5 APPLICATIONS TO PHYSICAL PROBLEMS 203
5.6 SUMMARY AND FURTHER READING 208 PROBLEMS 208 6. METHODS OF NONLINEAR
DYNAMICS I: PHASE PORTRAITS 212 6.1 A BRIEF SURVEY OF NONLINEAR
OPERATORS AND DIFFERENTIAL EQUATIONS 213 XVIII CONTENTS- 6.2 SOLUTION OF
NONLINEAR DIFFERENTIAL EQUATIONS: EXISTENCE AND UNIQUENESS THEOREMS 218
6.3 CRITICAL POINT ANALYSIS OF DIFFERENTIAL EQUATIONS 223 GENERALIZATION
TO THE CASE OF N VARIABLES 223 , DEFINITIONS: LINEAR SYSTEMS 224
DEFINITIONS (CONTINUED): A TWO DIMENSIONAL LINEAR SYSTEM 225 FURTHER
REMARKS ON LINEAR SYSTEMS 237 6.4 NONLINEAR SYSTEMS IN THE PLANE 238
LINEARIZATION AT A CRITICAL POINT 239 VOLTERRA-LOTKA SYSTEM 241 GENERAL
REMARKS 244 PROBLEMS 245 7. METHODS OF NONLINEAR DYNAMICS II: STABILITY
AND BIFURCATION 249 7.1 STABILITY OF CRITICAL POINTS AND LIAPUNOV
FUNCTIONS 249 STABILITY FOR NON-AUTONOMOUS SYSTEMS 249 STABILITY FOR
AUTONOMOUS SYSTEMS 253 LIAPUNOV FUNCTIONS 256 STABILITY AND LINEAR
APROXIMATION IN TWO DIMENSIONS 261 7.2 LIMIT CYCLE 264 7.3 INDEX OF A
CRITICAL POINT AND BENDIXSON CRITERION FOR PERIODIC SOLUTIONS 268 INDEX
OF A CRITICAL POINT 268 BENDIXSON S CRITERION FOR PERIODIC SOLUTIONS 270
7.4 BIFURCATION AND STRUCTURAL STABILITY 273 PHENOMENON OF BIFURCATION
273 ONE-DIMENSIONAL BIFURCATION 277 HOPF BIFURCATION 281 STRUCTURAL
STABILITY 283 CHAOS AND STANGE ATTRACTOR 284 7.5 APPLICATIONS TO
PHYSICAL PROBLEMS 287 CONSERVATIVE SYSTEMS 287 HAMILTONIAN SYSTEMS 294
7.6 SUMMARY AND FURTHER READING 297 PROBLEMS 297 8. SOME NONLINEAR
DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS 300 8.1 VAN DER POL EQUATION
300 LIENARD SYSTEMS AND VAB DER POL EQUATION 300 , DEPENDENCE OF THE
SOLUTION ON THE PARAMETER E 303 LARGE PARAMETER BEHAVIOUR OF THE
SOLUTION 304 8.2 SOLITARY-WAVE SOLUTIONS OF NONLINEAR DIFFERENTIAL
EQUATIONS 310 A BRIEF INTRODUCTION 310 KORTEWEG-DE VRIES (KDV) EQUATION
AND ITS SOLUTIONS 313 . SOME REMARKS ABOUT KDV AND MODIFIED KDV
EQUATIONS 325 -CONTENTS XIX 8.3 SOLITARY-WAVE SOLUTION OF NONLINEAR
SCHRODINGER EQUATION 326 8.4 APPLICATION TO PHYSICAL PROBLEMS 329
APPLICATIONS OF VAN DER POL EQUATION 329 APPLICTIONS OF KDV AND NLS
EQUATIONS 333 8.5 SUMMARY AND FURTHER READING 334 PROBLEMS 334 9. SOME
NONLINEAR INTEGRAL EQUATIONS AND THEIR SOLUTIONS 339 9.1 INVERSE
SCATTERING TRANSFORM METHOD 339 A BRIEF INTRODUCTION 339 TWO TYPICAL
EXAMPLES OF U(X) 342 INVERSE SCATTERING PROBLEM 346 CONNECTION WITH THE
KDV EQUATION 349 9.2 BACKLUND TRANSFORMATION AND THE SOLUTION OF KDV
EQUATION 352 9.3 THE LAX PAIR METHOD 357 9.4 HIROTA S METHOD OF BILINEAR
DERIVATIVES 361 9.5 PAINLEVE PROPERTY AND PAINLEVE TRANSCENDENTS 364 9.6
KADOMSTEV-PETVIASHVILI (KP) EQUATION AND ITS SOLUTIONS 367 9.7 SOLUTION
OF SOME NONLINEAR INTEGRAL EQUATIONS 368 EXISTENCE THEOREMS FOR NL
INTEGRAL EQUATIONS 368 SOME REPRESENTATIVE NONLINEAR INTEGRAL EQUATIONS
370 9.8 SUMMARY AND FURTHER READING 376 PROBLEMS 377 10. EXACT SOLUTION
OF SOME NONLINEAR DIFFERENTIAL EQUATIONS 378 10.1 RICCATI EQUATION 378
RICCATI EQUATION AND THE LINEAR DIFFERENTIAL EQUATION OF SECOND ORDER
379 SOLUTION OF THE ORIGINAL RICCATI EQUATION 381 FURTHER REMARKS ON
RICCATI EQUATION 383 10.2 EXACT SOLUTION OF SOME OTHER NLODES 385 10.3
NONLINEAR DIFFUSION EQUATIONS 387 CASE WHEN F(Q = C 388 CASE WHEN F(Q =
EXP (Q 391 10.4 EXACT SOLUTION OF SOME OTHER NLPDES 392 10.5
APPLICATIONS TO PHYSICAL PROBLEMS 396 CLASSICAL MECHANICS 396 QUANTUM
MECHANICS 396 ASTROPHYSICS 397 10.6 CONCLUDING DISCUSSION 399 PROBLEMS
400 XX CONTENTS- 11. SYMMETRIES OF DIFFERENTIAL EQUATIONS 402 11.1
SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS: AN INTRODUCTION 402
SYMMETRIES OF ALGEBRAIC EQUATIONS 402 GROUPS AND DIFFERENTIAL EQUATIONS
404 SOME BASIC RESULTS AND DEFINITIONS 405 11.2 EXTENDED TRANSFORMATIONS
OR PROLONGATIONS 419 11.3 EXTENDED INFINITESIMAL TRANSFORMATIONS 425
CASE OF ONE DEPENDENT AND ONE INDEPENDENT VARIABLES 426 CASE OF ONE
DEPENDENT AND P INDEPENDENT VARIABLES 426 11.4 INVARIANCE OF AN ORDINARY
DIFFERENTIAL EQUATION 427 11.5 INVARIANCE OF A PARTIAL DIFFERENTIAL
EQUATION 430 11.6 SYMMETRY GROUPS AND CONSERVATION LAWS 434 11.7
NOETHER S THEOREM AND LIE-BACKLUND SYMMETRIES 435 11.8 SUMMARY AND
FURTHER READING 440 PROBLEMS 441 12. NORMAL MODES IN NONLINEAR DYNAMICAL
SYSTEMS 443 12.1 NORMAL MODES OF LINEAR SYSTEMS: A BRIEF SURVEY 444 12.2
NORMAL MODES OF NONLINEAR SYSTEMS: A SIMPLE GENERALIZATION 448 12.3
TYPES OF MODE INTERACTIONS: A GROUP THEORETIC APPROACH 454 AN OVERVIEW
454 VARIOUS MODES AND THEIR INTERACTIONS 455 12.4 BUSHES OF MODES FOR A
DYNAMICAL SYSTEM 459 INTERACTIONS BETWEEN MODES OF AN N-PARTICLE
NONLINEAR SYSTEM 454 BUSH OF DYNAMICAL VARIABLES 461 12.5 DYNAMICAL BUSH
EQUATIONS: NORMAL FORMS THEORY 463 AN INTRODUCTION TO NORMAL FORM THEORY
463 INDICATOR OF A RESONANCE 465 INDICATOR OF RESONANCE FOR HAMILTONIAN
SYSTEMS 466 12.6 CONCLUDING DISCUSSION 469 APPENDIX A: SOME NUMERICAL
ASPECTS OF NONLINEAR DYNAMICAL SYSTEMS VIS-A-VIS CHAOS 470 AL.
INTRODUCTION 470 A2. PHASE FLOW AND MAPS 471 A3. CHARACTERIZATION OF THE
CHAOTIC MOTION 478 A4. ROUTES TO CHAOS 486 A5. FRACTALS AND HAUSDORFF
DIMENSION 493 A6. APPLICATION TO PHYSICAL PROBLEMS 495 A7. SUMMARY AND
FURTHER READING 497 -CONTENTS XXI APPENDIX B: DIFFERENTIATION AND
INTEGRATION UNDER THE INTEGRAL SIGN 499 BL. DIFFERENTIATION 499 B2.
INTEGRATION 501 B3. ALTERNATIVE DESCRIPTION OF DIFFERENTIATION UNDER THE
INTEGRAL SIGN 503 APPENDIX C: FURTHER REMARKS ON THE EXISTENCE OF LIMIT
CYCLES IN A LIENARD SYSTEM 506 BIBLIOGRAPHY 511 INDEX 514
|
| adam_txt |
ADVANCED METHODS OF MATH MATICAL PHYSICS SECOND EDITION ADVANCED METHODS
OF MATHEMATICAL PHYSICS SECOND EDITION R.S. KAUSHAL * D. PARASHAR ALPHA
SCIENCE INTERNATIONAL LTD. OXFORD, U.K. CONTENTS PREFACE TO THE SECOND
EDITION VII PREFACE TO THE FIRST EDITION IX ABBREVIATIONS, NOTATIONS AND
SYMBOLS XIII 1. GENERAL INTRODUCTION 1 2. THEORY OF FINITE GROUPS 5 2. 1
A BRIEF REVIEW OF SET THEORY 6 CARTESIAN PRODUCTS 6 MAPPINGS 7 BINARY
COMPOSITIONS 7 COMPOSITION TABLES 8 2.2 ABSTRACT GROUPS 8 PRELIMINARIES
8 CYCLIC GROUPS 10 PERMUTATION GROUP (S,,) 12 GROUPS OF SYMMETRY 14
CONJUGATE ELEMENTS AND CLASSES 18 SUBGROUPS 19 CENTER OF A GROUP 20
COSETS 20 NORMAL SUBGROUPS 22 FACTOR GROUPS 23 2.3 HOMOMORPHISMS 24
AUTOMORPHISMS 26 INNER AND OUTER AUTOMORPHISMS 27 GROUP OF AUTOMORPHISMS
27 DIRECT PRODUCT OF GROUPS 28 SEMIDIRECT PRODUCT OF GROUPS 29 2.4 GROUP
REPRESENTATIONS 30 INVARIANT SUBSPACES 32 REDUCIBLE REPRESENTATIONS 32
IRREDUCIBLE REPRESENTATIONS 34 UNITARY REPRESENTATIONS 36 SCHUR'S LEMMAS
37 THE ORTHOGONALITY THEOREM 39 CHARACTERS OF A REPRESENTATION 41 2.5
INTRODUCTION TO CONTINOUS GROUPS 42 LIE GROUPS 43 LIE GROUPS OF
TRANSFORMATIONS 44 INFINITE CONTINOUS' GROUPS 45 GENERATORS OF A LIE
GROUP 45 XVI CONTENTS- 2.6 APPLICATIONS TO PHYSICAL PROBLEMS 48
PERMUTATION GROUP (5,,) 48 UNITARY GROUP (SU(;I)) 57 SYMMETRY GROUP OF A
SQUARE 55 2.7 SUMMARY AND FURTHER READING 59 PROBLEMS 60 3. RUDIMENTS OF
TOPOLOGY AND DIFFERENTIAL GEOMETRY 63 3.1 PRELIMINARIES 63 DENUMERABLE
AND COUNTABLE SETS 64 LOWER AND UPPER BOUNDS 66 NEIGHBOURHOODS, OPEN AND
CLOSED SETS 66 CONTINUITY 68 LIMIT POINTS 69 BOLZANO-WEIERSTRASS THEOREM
70 ISOLATED, DENSE AND PERFECT SETS 71 3.2 METRIC SPACES 72 EUCLIDEAN
SPACE 73 HILBERT SPACE 74 DISTANCE BETWEEN SETS 74 OPEN AND CLOSED
SPHERES 75 EQUIVALENCE OF METRIC SPACES 76 3.3 TOPOLOGICAL SPACES 76
DEFINITION 76 UNION AND INTERSECTION OF TOPOLOGIES 78 LIMIT POINTS 79
CLOSURE OF A SET 79 INTERIOR, EXTERIOR AND BOUNDARY OF A SET 80 BASE FOR
A TOPOLOGY 82 HOUSDORFF SPACES 82 RELATIVE TOPOLOGIES 82 3.4 COMPACTNESS
83 SOME DEFINITIONS 83 HEINE-BOREL THEOREM 84 3.5 CONNECTEDNESS 86
SEPARATED SETS 86 CONNECTED SETS 86 CONNECTED SPACES 87 3.6 HOMOTOPY 88
HOMOTOPIC PATHS 88 SIMPLY CONNECTED SPACES 90 THE FUNDAMENTAL GROUP 91
3.7 ESSENTIALS OF DIFFERENTIA] GEOMETRY 94 SOME BASIC CONCEPTS AND
DEFINITIONS 94 DIFFERENTIABLE MANIFOLDS 96 DIFFEOMORPHISM 98 VECTOR
FIELDS 98 DIFFERENTIAL FORMS 101 3.8 SUMMARY AND FURTHER READING 103
PROBLEMS 104 -CONTENTS XVII 4. INTEGRAL EQUATIONS, STURM-LIOUVILLE
THEORY AND GREEN'S FUNCTIONS 107 4.1 TERMINOLOGY AND DEFINITIONS 707'
FREDHOLM INTEGRAL EQUATIONS 108 ' VOLTERRA INTEGRAL EQUATIONS 109
DIFFERENTIATION OF A FUNCTION UNDER THE INTEGRAL SIGN 110 RELATION
BETWEEN DIFFERENTIAL AND INTEGRAL EQUATIONS 111 4.2 SOLUTION OF INTEGRAL
EQUATIONS 113 THE LIOUVILLE-NEUMANN SERIES METHOD 113 THE FREDHOLM
METHOD 124 THE HILBERT-SCHMIDT THEORY 134 4.3 STURM-LIOUVILLE THEORY 142
ADJOINT DIFFERENTIAL SYTSTEM 143 THE STURM-LIOUVILLE PROBLEMS:
EIGENVALUES AND EIGENFUNCTIONS 145 4.4 THE GREEN'S FUNCTIONS 1.48 ,
DETERMINATION OF G(X, T) 149 CONNECTION WITH INHOMOGENEOUS
STURM-LIOUVILLE EQUATION 152 4.5 APPLICTIONS TO PHYSICS PROBLEMS 153 THE
INFLUENCE FUNCTION 154 * . * THE ABEL'S INTEGRAL EQUATION 156 ' * * ' *'
: *'***'* 4.6 SUMMARY AND FURTHER READING 158 PROBLEMS 159 ' , 5.
STOCHASTIC PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS ,.,,.* , .
162 5.1 RANDOM VARIABLE AND DISTRIBUTION FUNCTION OF RANDOM VARIABLES
163 , * SOME BASIC DEFINITIONS AND RESULTS 163 MULTIDIMENSIONAL
DISTRIBUTION FUNCTIONS 168 FUNCTIONS OF RANDOM VARIABLES 171 , THE
STIELTJES INTEGRAL 174 5.2 NUMERICAL CHARACTERSTICS OF RANDOM VARIABLES:
MOMENTS OF THE DISTRIBUTION FUNCTION 176 MATHEMATICAL EXPECTATION, 176 :
. . . . . * * VARIANCE 178 CO VARIANCE AND CO VARIANCE MATRIX 182
CHARACTERISTIC FUNCTION OF RANDOM VARIABLES 185 5.3 STOCHASTIC
PROCESSES: MARKOV PROCESS 186 CLASSIFICATION OF STOCHASTIC PROCESSES 187
MARKOV PROCESSES: FOKKER-PLANCK EQUATION 191 GENERAL THEORY OF
CONTINUOUS (MARKOV) PROCESSES 193 , . . 5.4 STOCHASTIC DIFFERENTIAL
EQUATIONS: AN INTRODUCTION 799 5.5 APPLICATIONS TO PHYSICAL PROBLEMS 203
5.6 SUMMARY AND FURTHER READING 208 PROBLEMS 208 6. METHODS OF NONLINEAR
DYNAMICS I: PHASE PORTRAITS 212 6.1 A BRIEF SURVEY OF NONLINEAR
OPERATORS AND DIFFERENTIAL EQUATIONS 213 XVIII CONTENTS- 6.2 SOLUTION OF
NONLINEAR DIFFERENTIAL EQUATIONS: EXISTENCE AND UNIQUENESS THEOREMS 218
6.3 CRITICAL POINT ANALYSIS OF DIFFERENTIAL EQUATIONS 223 GENERALIZATION
TO THE CASE OF N VARIABLES 223 , DEFINITIONS: LINEAR SYSTEMS 224
DEFINITIONS (CONTINUED): A TWO DIMENSIONAL LINEAR SYSTEM 225 FURTHER
REMARKS ON LINEAR SYSTEMS 237 6.4 NONLINEAR SYSTEMS IN THE PLANE 238
LINEARIZATION AT A CRITICAL POINT 239 VOLTERRA-LOTKA SYSTEM 241 GENERAL
REMARKS 244 PROBLEMS 245 7. METHODS OF NONLINEAR DYNAMICS II: STABILITY
AND BIFURCATION 249 7.1 STABILITY OF CRITICAL POINTS AND LIAPUNOV
FUNCTIONS 249 STABILITY FOR NON-AUTONOMOUS SYSTEMS 249 STABILITY FOR
AUTONOMOUS SYSTEMS 253 LIAPUNOV FUNCTIONS 256 STABILITY AND LINEAR
APROXIMATION IN TWO DIMENSIONS 261 7.2 LIMIT CYCLE 264 7.3 INDEX OF A
CRITICAL POINT AND BENDIXSON CRITERION FOR PERIODIC SOLUTIONS 268 INDEX
OF A CRITICAL POINT 268 BENDIXSON'S CRITERION FOR PERIODIC SOLUTIONS 270
7.4 BIFURCATION AND STRUCTURAL STABILITY 273 PHENOMENON OF BIFURCATION
273 ONE-DIMENSIONAL BIFURCATION 277 HOPF BIFURCATION 281 STRUCTURAL
STABILITY 283 CHAOS AND STANGE ATTRACTOR 284 7.5 APPLICATIONS TO
PHYSICAL PROBLEMS 287 CONSERVATIVE SYSTEMS 287 HAMILTONIAN SYSTEMS 294
7.6 SUMMARY AND FURTHER READING 297 PROBLEMS 297 8. SOME NONLINEAR
DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS 300 8.1 VAN DER POL EQUATION
300 LIENARD SYSTEMS AND VAB DER POL EQUATION 300 , DEPENDENCE OF THE
SOLUTION ON THE PARAMETER E 303 LARGE PARAMETER BEHAVIOUR OF THE
SOLUTION 304 8.2 SOLITARY-WAVE SOLUTIONS OF NONLINEAR DIFFERENTIAL
EQUATIONS 310 A BRIEF INTRODUCTION 310 KORTEWEG-DE VRIES (KDV) EQUATION
AND ITS SOLUTIONS 313 . SOME REMARKS ABOUT KDV AND MODIFIED KDV
EQUATIONS 325 -CONTENTS XIX 8.3 SOLITARY-WAVE SOLUTION OF NONLINEAR
SCHRODINGER EQUATION 326 8.4 APPLICATION TO PHYSICAL PROBLEMS 329
APPLICATIONS OF VAN DER POL EQUATION 329 APPLICTIONS OF KDV AND NLS
EQUATIONS 333 8.5 SUMMARY AND FURTHER READING 334 PROBLEMS 334 9. SOME
NONLINEAR INTEGRAL EQUATIONS AND THEIR SOLUTIONS 339 9.1 INVERSE
SCATTERING TRANSFORM METHOD 339 A BRIEF INTRODUCTION 339 TWO TYPICAL
EXAMPLES OF U(X) 342 INVERSE SCATTERING PROBLEM 346 CONNECTION WITH THE
KDV EQUATION 349 9.2 BACKLUND TRANSFORMATION AND THE SOLUTION OF KDV
EQUATION 352 9.3 THE LAX PAIR METHOD 357 9.4 HIROTA'S METHOD OF BILINEAR
DERIVATIVES 361 9.5 PAINLEVE PROPERTY AND PAINLEVE TRANSCENDENTS 364 9.6
KADOMSTEV-PETVIASHVILI (KP) EQUATION AND ITS SOLUTIONS 367 9.7 SOLUTION
OF SOME NONLINEAR INTEGRAL EQUATIONS 368 EXISTENCE THEOREMS FOR NL
INTEGRAL EQUATIONS 368 SOME REPRESENTATIVE NONLINEAR INTEGRAL EQUATIONS
370 9.8 SUMMARY AND FURTHER READING 376 PROBLEMS 377 10. EXACT SOLUTION
OF SOME NONLINEAR DIFFERENTIAL EQUATIONS 378 10.1 RICCATI EQUATION 378
RICCATI EQUATION AND THE LINEAR DIFFERENTIAL EQUATION OF SECOND ORDER
379 SOLUTION OF THE ORIGINAL RICCATI EQUATION 381 FURTHER REMARKS ON
RICCATI EQUATION 383 10.2 EXACT SOLUTION OF SOME OTHER NLODES 385 10.3
NONLINEAR DIFFUSION EQUATIONS 387 CASE WHEN F(Q = C 388 CASE WHEN F(Q =
EXP (Q 391 10.4 EXACT SOLUTION OF SOME OTHER NLPDES 392 10.5
APPLICATIONS TO PHYSICAL PROBLEMS 396 CLASSICAL MECHANICS 396 QUANTUM
MECHANICS 396 ASTROPHYSICS 397 10.6 CONCLUDING DISCUSSION 399 PROBLEMS
400 XX CONTENTS- 11. SYMMETRIES OF DIFFERENTIAL EQUATIONS 402 11.1
SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS: AN INTRODUCTION 402
SYMMETRIES OF ALGEBRAIC EQUATIONS 402 GROUPS AND DIFFERENTIAL EQUATIONS
404 SOME BASIC RESULTS AND DEFINITIONS 405 11.2 EXTENDED TRANSFORMATIONS
OR PROLONGATIONS 419 11.3 EXTENDED INFINITESIMAL TRANSFORMATIONS 425
CASE OF ONE DEPENDENT AND ONE INDEPENDENT VARIABLES 426 CASE OF ONE
DEPENDENT AND P INDEPENDENT VARIABLES 426 11.4 INVARIANCE OF AN ORDINARY
DIFFERENTIAL EQUATION 427 11.5 INVARIANCE OF A PARTIAL DIFFERENTIAL
EQUATION 430 11.6 SYMMETRY GROUPS AND CONSERVATION LAWS 434 11.7
NOETHER'S THEOREM AND LIE-BACKLUND SYMMETRIES 435 11.8 SUMMARY AND
FURTHER READING 440 PROBLEMS 441 12. NORMAL MODES IN NONLINEAR DYNAMICAL
SYSTEMS 443 12.1 NORMAL MODES OF LINEAR SYSTEMS: A BRIEF SURVEY 444 12.2
NORMAL MODES OF NONLINEAR SYSTEMS: A SIMPLE GENERALIZATION 448 12.3
TYPES OF MODE INTERACTIONS: A GROUP THEORETIC APPROACH 454 AN OVERVIEW
454 VARIOUS MODES AND THEIR INTERACTIONS 455 12.4 BUSHES OF MODES FOR A
DYNAMICAL SYSTEM 459 INTERACTIONS BETWEEN MODES OF AN N-PARTICLE
NONLINEAR SYSTEM 454 BUSH OF DYNAMICAL VARIABLES 461 12.5 DYNAMICAL BUSH
EQUATIONS: NORMAL FORMS THEORY 463 AN INTRODUCTION TO NORMAL FORM THEORY
463 INDICATOR OF A RESONANCE 465 INDICATOR OF RESONANCE FOR HAMILTONIAN
SYSTEMS 466 12.6 CONCLUDING DISCUSSION 469 APPENDIX A: SOME NUMERICAL
ASPECTS OF NONLINEAR DYNAMICAL SYSTEMS VIS-A-VIS CHAOS 470 AL.
INTRODUCTION 470 A2. PHASE FLOW AND MAPS 471 A3. CHARACTERIZATION OF THE
CHAOTIC MOTION 478 A4. ROUTES TO CHAOS 486 A5. FRACTALS AND HAUSDORFF
DIMENSION 493 A6. APPLICATION TO PHYSICAL PROBLEMS 495 A7. SUMMARY AND
FURTHER READING 497 -CONTENTS XXI APPENDIX B: DIFFERENTIATION AND
INTEGRATION UNDER THE INTEGRAL SIGN 499 BL. DIFFERENTIATION 499 B2.
INTEGRATION 501 B3. ALTERNATIVE DESCRIPTION OF DIFFERENTIATION UNDER THE
INTEGRAL SIGN 503 APPENDIX C: FURTHER REMARKS ON THE EXISTENCE OF LIMIT
CYCLES IN A LIENARD SYSTEM 506 BIBLIOGRAPHY 511 INDEX 514 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Kaushal, Radhey Shyam 1944- Parashar, D. |
| author_GND | (DE-588)10434458X |
| author_facet | Kaushal, Radhey Shyam 1944- Parashar, D. |
| author_role | aut aut |
| author_sort | Kaushal, Radhey Shyam 1944- |
| author_variant | r s k rs rsk d p dp |
| building | Verbundindex |
| bvnumber | BV023367709 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20 |
| callnumber-search | QC20 |
| callnumber-sort | QC 220 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 950 |
| classification_tum | PHY 011f |
| ctrlnum | (OCoLC)221682101 (DE-599)BVBBV023367709 |
| dewey-full | 530.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV023367709 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:11:23Z |
| indexdate | 2024-07-09T21:16:59Z |
| institution | BVB |
| isbn | 9781842654095 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016551032 |
| oclc_num | 221682101 |
| open_access_boolean | |
| owner | DE-20 DE-19 DE-BY-UBM DE-91G DE-BY-TUM |
| owner_facet | DE-20 DE-19 DE-BY-UBM DE-91G DE-BY-TUM |
| physical | XXI, 519 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Alpha Science Internat. |
| record_format | marc |
| spelling | Kaushal, Radhey Shyam 1944- Verfasser (DE-588)10434458X aut Advanced methods of mathematical physics R. S. Kaushal ; D. Parashar Mathematical physics 2. ed. Oxford Alpha Science Internat. 2008 XXI, 519 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s DE-604 Parashar, D. Verfasser aut HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016551032&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kaushal, Radhey Shyam 1944- Parashar, D. Advanced methods of mathematical physics Mathematische Physik Mathematical physics Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4037952-8 |
| title | Advanced methods of mathematical physics |
| title_alt | Mathematical physics |
| title_auth | Advanced methods of mathematical physics |
| title_exact_search | Advanced methods of mathematical physics |
| title_exact_search_txtP | Advanced methods of mathematical physics |
| title_full | Advanced methods of mathematical physics R. S. Kaushal ; D. Parashar |
| title_fullStr | Advanced methods of mathematical physics R. S. Kaushal ; D. Parashar |
| title_full_unstemmed | Advanced methods of mathematical physics R. S. Kaushal ; D. Parashar |
| title_short | Advanced methods of mathematical physics |
| title_sort | advanced methods of mathematical physics |
| topic | Mathematische Physik Mathematical physics Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik Mathematical physics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016551032&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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