Methods for solving mathematical physics problems:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Internat. Science Publ.
2006
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| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 320 S. |
| ISBN: | 9781904602057 1904602053 |
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| 100 | 1 | |a Agoškov, Vasilij I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Methods for solving mathematical physics problems |c V. I. Agoshkov ; P. B. Dubovski ; V. P. Shutyaev |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge Internat. Science Publ. |c 2006 | |
| 300 | |a XIV, 320 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathematical physics - Methodology | |
| 650 | 4 | |a Mathematische Physik - Mathematische Methode | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Mathematical physics |x Methodology | |
| 650 | 0 | 7 | |a Mathematische Methode |0 (DE-588)4155620-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Mathematische Methode |0 (DE-588)4155620-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Dubovski, Pavel B. |e Verfasser |4 aut | |
| 700 | 1 | |a Šutjaev, Viktor P. |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m GBV Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016438722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| adam_text | METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS V.I. AGOSHKOV, P.B.
DUBOVSKI, V.P. SHUTYAEV CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING
CONTENTS PREFACE 1. MAIN PROBLEMS OF MATHEMATICAL PHYSICS 1 MAIN
CONCEPTS AND NOTATIONS 1 1. INTRODUCTION 2 2. CONCEPTS AND ASSUMPTIONS
FROM THE THEORY OF FUNCTIONS AND FUNCTIONAL ANALYSIS _ 3 2.1. POINT
SETS. CLASS OF FUNCTIONS C P (Q), C P (Q) 3 2.1.1. POINT SETS *. 3
2.1.2. CLASSES C(Q),(?(Q) 4 2.2. EXAMPLES FROM THE THEORY OF LINEAR
SPACES 5 2.2.1. NORMALISED SPACE 5 2.2.2. THE SPACE OF CONTINUOUS
FUNCTIONS C(Q) 6 2.2.3. SPACES C*(FI) 6 2.2.4. SPACE L P (Q) 7 2.3. L 2
(Q) SPACE. ORTHONORMAL SYSTEMS 9 2.3.1. HILBERT SPACES 9 2.3.2. SPACE
L,(FI) 11 2.3.3. ORTHONORMAL SYSTEMS 11 2.4. LINEAR OPERATORS AND
FUNCTIONAL 13 2.4.1. LINEAR OPERATORS AND FUNCTIONALS 13 2.4.2. INVERSE
OPERATORS 15 2.4.3. ADJOINT, SYMMETRIC AND SELF-ADJOINT OPERATORS 15
2.4.4. POSITIVE OPERATORS AND ENERGETIC SPACE 16 2.4.5. LINEAR
EQUATIONS ;. 17 2.4.6. EIGENVALUE PROBLEMS . 17 2.5. GENERALIZED
DERIVATIVES. SOBOLEV SPACES 19 2.5.1. GENERALIZED DERIVATIVES 19 2.5.2.
SOBOLEV SPACES 20 2.5.3. THE GREEN FORMULA . 21 3. MAIN EQUATIONS AND
PROBLEMS OF MATHEMATICAL PHYSICS 22 3.1. MAIN EQUATIONS OF MATHEMATICAL
PHYSICS 22 3.1.1. LAPLACE AND POISSON EQUATIONS 23 .2. EQUATIONS OF
OSCILLATIONS 24 .3. HELMHOLTZ EQUATION 26 .4. DIFFUSION AND HEAT
CONDUCTION EQUATIONS 26 .5. MAXWELL AND TELEGRAPH EQUATIONS 27 .6.
TRANSFER EQUATION 28 .7. GAS- AND HYDRODYNAMIC EQUATIONS : 29 .8.
CLASSIFICATION OF LINEAR DIFFERENTIAL EQUATIONS 29 VII 3.2. FORMULATION
OF THE MAIN PROBLEMS OF MATHEMATICAL PHYSICS 32 3.2.1. CLASSIFICATION OF
BOUNDARY-VALUE PROBLEMS 32 3.2.2. THE CAUCHY PROBLEM 33 3.2.3. THE
BOUNDARY-VALUE PROBLEM FOR THE ELLIPTICAL EQUATION 34 3.2.4. MIXED
PROBLEMS 35 3.2.5. VALIDITY OF FORMULATION OF PROBLEMS.
CAUCHY-KOVALEVSKII THEOREM 35 3.3. GENERALIZED FORMULATIONS AND
SOLUTIONS OF MATHEMATICAL PHYSICS PROBLEMS... 37 3.3.1. GENERALIZED
FORMULATIONS AND SOLUTIONS OF ELLIPTICAL PROBLEMS 38 3.3.2. GENERALIZED
FORMULATIONS AND SOLUTION OF HYPERBOLIC PROBLEMS 41 3.3.3. THE
GENERALIZED FORMULATION AND SOLUTIONS OF PARABOLIC PROBLEMS 43 3.4.
VARIATIONAL FORMULATIONS OF PROBLEMS 45 3.4.1. VARIATIONAL FORMULATION
OF PROBLEMS IN THE CASE OF POSITIVE DEFINITE OPERATORS 45 3.4.2.
VARIATIONAL FORMULATION OF THE PROBLEM IN THE CASE OF POSITIVE OPERATORS
. 46 3.4.3. VARIATIONAL FORMULATION OF THE BASIC ELLIPTICAL PROBLEMS 47
3.5. INTEGRAL EQUATIONS 49 3.5.1. INTEGRAL FREDHOLM EQUATION OF THE 1ST
AND 2ND KIND 49 3.5.2. VOLTERRA INTEGRAL EQUATIONS 50 3.5.3. INTEGRAL
EQUATIONS WITH A POLAR KERNEL 51 3.5.4. FREDHOLM THEOREM 51 3.5.5.
INTEGRAL EQUATION WITH THE HERMITIAN KERNEL 52 BIBLIOGRAPHIC COMMENTARY
54 2. METHODS OF POTENTIAL THEORY 56 MAIN CONCEPTS AND DESIGNATIONS H 56
1. INTRODUCTION 57 2. FUNDAMENTALS OF POTENTIAL THEORY 58 2.1.
ADDITIONAL INFORMATION FROM MATHEMATICAL ANALYSIS 58 2.1.1 MAIN
ORTHOGONAL COORDINATES 58 2.1.2. MAIN DIFFERENTIAL OPERATIONS OF THE
VECTOR FIELD 58 2.1.3. FORMULAE FROM THE FIELD THEORY 59 2.1.4. MAIN
PROPERTIES OF HARMONIC FUNCTIONS 69 2.2 POTENTIAL OF VOLUME MASSES OR
CHARGES 61 2.2.1. NEWTON (COULOMB) POTENTIAL 61 2.2.2. THE PROPERTIES OF
THE NEWTON POTENTIAL 61 2.2.3. POTENTIAL OF A HOMOGENEOUS SPHERE 62
2.2.4. PROPERTIES OF THE POTENTIAL OF VOLUME-DISTRIBUTED MASSES 62 2.3.
LOGARITHMIC POTENTIAL 63 2.3.1. DEFINITION OF THE LOGARITHMIC POTENTIAL
63 2.3.2. THE PROPERTIES OF THE LOGARITHMIC POTENTIAL 63 2.3.3. THE
LOGARITHMIC POTENTIAL OF A CIRCLE WITH CONSTANT DENSITY 64 2.4. THE
SIMPLE LAYER POTENTIAL 64 2.4.1. DEFINITION OF THE SIMPLE LAYER
POTENTIAL IN SPACE 64 2.4.2. THE PROPERTIES OF THE SIMPLE LAYER
POTENTIAL 65 2.4.3. THE POTENTIAL OF THE HOMOGENEOUS SPHERE 66 2.4.4.
THE SIMPLE LAYER POTENTIAL ON A PLANE 66 VNI 2.5. DOUBLE LAYER POTENTIAL
67 2.5.1. DIPOLE POTENTIAL 67 2.5.2. THE DOUBLE LAYER POTENTIAL IN SPACE
AND ITS PROPERTIES 67 2.5.3. THE LOGARITHMIC DOUBLE LAYER POTENTIAL AND
ITS PROPERTIES 69 3. USING THE POTENTIAL THEORY IN CLASSIC PROBLEMS OF
MATHEMATICAL PHYSICS 70 3.1. SOLUTION OF THE LAPLACE AND POISSON
EQUATIONS 70 3.1.1. FORMULATION OF THE BOUNDARY-VALUE PROBLEMS OF THE
LAPLACE EQUATION ....70 3.1.2 SOLUTION OF THE DIRICHLET PROBLEM IN SPACE
71 3.1.3. SOLUTION OF THE DIRICHLET PROBLEM ON A PLANE 72 3.1.4.
SOLUTION OF THE NEUMANN PROBLEM 73 3.1.5. SOLUTION OF THE THIRD
BOUNDARY-VALUE PROBLEM FOR THE LAPLACE EQUATION ..74 3.1.6. SOLUTION OF
THE BOUNDARY-VALUE PROBLEM FOR THE POISSON EQUATION 75 3.2. THE GREEN
FUNCTION OF THE LAPLACE OPERATOR 76 3.2.1. THE POISSON EQUATION 76
3.2.2. THE GREEN FUNCTION 76 3.2.3. SOLUTION OF THE DIRICHLET PROBLEM
FOR SIMPLE DOMAINS 77 3.3 SOLUTION OF THE LAPLACE EQUATION FOR COMPLEX
DOMAINS 78 3.3.1. SCHWARZ METHOD 78 3.3.2. THE SWEEP METHOD 80 4. OTHER
APPLICATIONS OF THE POTENTIAL METHOD 81 4.1. APPLICATION OF THE
POTENTIAL METHODS TO THE HELMHOLTZ EQUATION 81 4.1.1. MAIN FACTS ,. 81
4.1.2. BOUNDARY-VALUE PROBLEMS FOR THE HELMHOLTZ EQUATIONS 82 4.1.3.
GREEN FUNCTION 84 4.1.4. EQUATION AV-XV - 0 85 4.2. NON-STATIONARY
POTENTIALS ; 86 4.2.1 POTENTIALS FOR THE ONE-DIMENSIONAL HEAT EQUATION
86 4.2.2. HEAT SOURCES IN MULTIDIMENSIONAL CASE 88 4.2.3. THE
BOUNDARY-VALUE PROBLEM FOR THE WAVE EQUATION 90 BIBLIOGRAPHIC COMMENTARY
92, 3. EIGENFUNCTION METHODS 1 94 MAIN CONCEPTS AND NOTATIONS .: 94 1.
INTRODUCTION 94 2. EIGENVALUE PROBLEMS 95 2.1. FORMULATION AND THEORY 95
2.2. EIGENVALUE PROBLEMS FOR DIFFERENTIAL OPERATORS 98 2.3. PROPERTIES
OF EIGENVALUES AND EIGENFUNCTIONS 99 2.4. FOURIER SERIES 100 2.5.
EIGENFUNCTIONS OF SOME ONE-DIMENSIONAL PROBLEMS 102 3. SPECIAL FUNCTIONS
103 3.1. SPHERICAL FUNCTIONS 103 3.2. LEGENDRE POLYNOMIALS 105 3.3.
CYLINDRICAL FUNCTIONS 106 3.4. CHEBYSHEF, LAGUERRE AND HERMITE
POLYNOMIALS 107 3.5. MATHIEU FUNCTIONS AND HYPERGEOMETRICAL FUNCTIONS
109 4. EIGENFUNCTION METHOD 110 4.1. GENERAL SCHEME OF THE EIGENFUNCTION
METHOD .* 110 4.2. THE EIGENFUNCTION METHOD FOR DIFFERENTIAL EQUATIONS
OF MATHEMATICAL PHYSICS ILL 4.3. SOLUTION OF PROBLEMS WITH
NONHOMOGENEOUS BOUNDARY CONDITIONS 114 5. EIGENFUNCTION METHOD FOR
PROBLEMS OF THE THEORY OF ELECTROMAGNETIC PHENOMENA 115 5.1. THE PROBLEM
OF ABOUNDED TELEGRAPH LINE 115 5.2. ELECTROSTATIC FIELD INSIDE AN
INFINITE PRISM 117 5.3. PROBLEM OF THE ELECTROSTATIC FIELD INSIDE A
CYLINDER 117 5.4. THE FIELD INSIDE A BALL AT A GIVEN POTENTIAL ON ITS
SURFACE 118 5.5 THE FIELD OF A CHARGE INDUCED ON A BALL 120. 6.
EIGENFUNCTION METHOD FOR HEAT CONDUCTIVITY PROBLEMS 121 6.1. HEAT
CONDUCTIVITY IN A BOUNDED BAR 121 6.2. STATIONARY DISTRIBUTION OF
TEMPERATURE IN AN INFINITE PRISM 122 6.3. TEMPERATURE DISTRIBUTION OF A
HOMOGENEOUS CYLINDER 123 7. EIGENFUNCTION METHOD FOR PROBLEMS IN THE
THEORY OF OSCILLATIONS 124 7.1. FREE OSCILLATIONS OF A HOMOGENEOUS
STRING 124 7.2. OSCILLATIONS OF THE STRING WITH A MOVING END 125 7.3.
PROBLEM OF ACOUSTICS OF FREE OSCILLATIONS OF GAS 126 7.4. OSCILLATIONS
OF A MEMBRANE WITH A FIXED END 127 7.5. PROBLEM OF OSCILLATION OF A
CIRCULAR MEMBRANE 128 BIBLIOGRAPHIC COMMENTARY 129 4. METHODS OFJNTEGRAL
TRANSFORMS 130 MAIN CONCEPTS AND DEFINITIONS 130 1. INTRODUCTION 131 2.
MAIN INTEGRAL TRANSFORMATIONS .: 132 2.1. FOURIER TRANSFORM 132 2.1.1.
THE MAIN PROPERTIES OF FOURIER TRANSFORMS 133 2.1.2. MULTIPLE FOURIER
TRANSFORM 134 2.2. LAPLACE TRANSFORM !*. 134 2.2.1. LAPLACE INTEGRAL .
134 2.2.2. THE INVERSION FORMULA FOR THE LAPLACE TRANSFORM 135 2.2.3.
MAIN FORMULAE AND LIMITING THEOREMS 135 2.3. MELLIN TRANSFORM 135 2.4.
HANKEL TRANSFORM 136 2.5. MEYER TRANSFORM 138 2.6. KONTOROVICH-LEBEDEV
TRANSFORM 138 2.7. MELLER-FOCK TRANSFORM 139 2.8 HILBERT TRANSFORM 140
2.9. LAGUERRE AND LEGENDRE TRANSFORMS 140 2.10 BOCHNER AND CONVOLUTION
TRANSFORMS, WAVELETS AND CHAIN TRANSFORMS 141 3. USING INTEGRAL
TRANSFORMS IN PROBLEMS OF OSCILLATION THEORY 143 3.1. ELECTRICAL
OSCILLATIONS 143 3.2. TRANSVERSE VIBRATIONS OF A STRING 143 3.3.
TRANSVERSE VIBRATIONS OF AN INFINITE CIRCULAR MEMBRANE 146 4. USING
INTEGRAL TRANSFORMS IN HEAT CONDUCTIVITY PROBLEMS 147 4.1. SOLVING HEAT
CONDUCTIVITY PROBLEMS USING THE LAPLACE TRANSFORM 147 4.2. SOLUTION OF A
HEAT CONDUCTIVITY PROBLEM USING FOURIER TRANSFORMS 148 4.3. TEMPERATURE
REGIME OF A SPHERICAL BALL 149 5. USING INTEGRAL TRANSFORMATIONS IN THE
THEORY OF NEUTRON DIFFUSION 149 5.1. THE SOLUTION OF THE EQUATION OF
DECELERATION OF NEUTRONS FOR A MODERATOR OF INFINITE DIMENSIONS 150 5.2.
THE PROBLEM OF DIFFUSION OF THERMAL NEUTRONS 150 6. APPLICATION OF
INTEGRAL TRANSFORMATIONS TO HYDRODYNAMIC PROBLEMS 151 6.1. A
TWO-DIMENSIONAL VORTEX-FREE FLOW OF AN IDEAL LIQUID 151 6.2. THE FLOW OF
THE IDEAL LIQUID THROUGH A SLIT 152 6.3. DISCHARGE OF THE IDEAL LIQUID
THROUGH A CIRCULAR ORIFICE 153 7. USING INTEGRAL TRANSFORMS IN
ELASTICITY THEORY 155 7.1. AXISYMMETRIC STRESSES IN ACYLINDER 155 7.2.
BUSSINESQ PROBLEM;FOR THE HALF SPACE 157 7.3. DETERMINATION OF STRESSES
IN A WEDGE 158 8. USING INTEGRAL TRANSFORMS IN COAGULATION KINETICS 159
8.1. EXACT SOLUTION OF THE COAGULATION EQUATION 159 8.2. VIOLATION OF
THE MASS CONSERVATION LAW 161 BIBLIOGRAPHIC COMMENTARY .; 162 5. METHODS
OF DISCRETISATION OF MATHEMATICAL PHYSICS PROBLEMS 163 MAIN DEFINITIONS
AND NOTATIONS 163 1. INTRODUCTION . 164 2. FINITE-DIFFERENCE METHODS
166 2.1. THE NET METHOD 166 2.1.1. MAIN CONCEPTS AND DEFINITIONS OF THE
METHOD 166 2.1.2. GENERAL DEFINITIONS OF THE NET METHOD. THE CONVERGENCE
THEOREM 170, 2.1.3. THE NET METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
173 2.2. THE METHOD OF ARBITRARY LINES :. 182 2.2.1. THE METHOD OF
ARBITRARY LINES FOR PARABOLIC-TYPE EQUATIONS 182 2.2.2. THE METHOD OF
ARBITRARY LINES FOR HYPERBOLIC EQUATIONS 184 2.2.3. THE METHOD OF
ARBITRARY LINES FOR ELLIPTICAL EQUATIONS 185 2.3. THE NET METHOD FOR
INTEGRAL EQUATIONS (THE QUADRATURE METHOD) 187 3. VARIATIONAL METHODS
188 3.1. MAIN CONCEPTS OF VARIATIONAL FORMULATIONS OF PROBLEMS AND
VARIATIONAL METHODS .....188 3.1.1. VARIATIONAL FORMULATIONS OF PROBLEMS
188 3.1.2. CONCEPTS OF THE DIRECT METHODS IN CALCULUS OF VARIATIONS 189
3.2. THE RITZ METHOD 190 3.2.1. THE CLASSIC RITZ METHOD 190 3.2.2. THE
RITZ METHOD IN ENERGY SPACES 192 3.2.3. NATURAL AND MAIN BOUNDARY-VALUE
CONDITIONS 194 3.3. THE METHOD OF LEAST SQUARES 195 XI 3.4. KANTOROVICH,
COURANT AND TREFFTZ METHODS 196 3.4.1. THE KANTOROVICH METHOD 1% 3.4.2.
COURANT METHOD 1% 3.4.3. TREFFTZ METHOD , 197 3.5. VARIATIONAL METHODS
IN THE EIGENVALUE PROBLEM 199 4. PROJECTION METHODS 201 4.1.
THEBUBNOV-GALERKIN METHOD V 201 4.1.1. THEBUBNOV-GALERKIN METHOD (A
GENERAL CASE) 201 4.1.2 THEBUBNOV-GALERKIN METHOD (I4= 0 +J5) 202 4.2.
THE MOMENTS METHOD 204 4.3. PROJECTION METHODS IN THE HILBERT AND BANACH
SPACES 205 4.3.1. THE PROJECTION METHOD IN THE HILBERT SPACE 205 4.3.2.
THE GALERKIN-PETROV METHOD 206 4.3.3. THE PROJECTION METHOD IN THE
BANACH SPACE 206 4.3.4. THE COLLOCATION METHOD 208 4.4. MAIN CONCEPTS OF
THE PROJECTION-GRID METHODS 208 5. METHODS OF INTEGRAL IDENTITIES 210
5.1. THE MAIN CONCEPTS OF THE METHOD 210 5.2. THE METHOD OF MARCHUK S
INTEGRAL IDENTITY 211 5.3. GENERALIZED FORMULATION OF THE METHOD OF
INTEGRAL IDENTITIES 213 5.3.1. ALGORITHM OF CONSTRUCTING INTEGRAL
IDENTITIES 213 5.3.2. THE DIFFERENCE METHOD OF APPROXIMATING THE
INTEGRAL IDENTITIES 214 5.3.3. THE PROJECTION METHOD OF APPROXIMATING
THE INTEGRAL IDENTITIES 215 5.4. APPLICATIONS OF THE METHODS OF INTEGRAL
IDENTITIES IN MATHEMATICAL PHYSICS PROBLEMS 217 5.4.1. THE METHOD OF
INTEGRAL IDENTITIES FOR THE DIFFUSION EQUATION 217 5.4.2. THE SOLUTION
OF DEGENERATING EQUATIONS 219 5.4.3. THE METHOD OF INTEGRAL IDENTITIES
FOR EIGENVALUE PROBLEMS 221 BIBLIOGRAPHIC COMMENTARY 223 6. SPLITTING
METHODS 224 1. INTRODUCTION 224 2. INFORMATION FROM THE THEORY OF
EVOLUTION EQUATIONS AND DIFFERENCE SCHEMES . 225 2.1. EVOLUTION
EQUATIONS 225 2.1.1. THE CAUCHY PROBLEM 225 2.1.2. THE NONHOMOGENEOUS
EVOLUTION EQUATION 228 2.1.3. EVOLUTION EQUATIONS WITH BOUNDED OPERATORS
229 2.2. OPERATOR EQUATIONS IN FINITE-DIMENSIONAL SPACES 231 2.2.1. THE
EVOLUTION SYSTEM 231 2.2.2. STATIONARISATION METHOD 232 2.3. CONCEPTS
AND INFORMATION FROM THE THEORY OF DIFFERENCE SCHEMES 233 2.3.1.
APPROXIMATION 233 2.3.2. STABILITY 239 2.3.3. CONVERGENCE : 240 2.3.4.
THE SWEEP METHOD 241 3. SPLITTING METHODS ..: 242 XII 3.1. THE METHOD OF
COMPONENT SPLITTING (THE FRACTIONAL STEP METHODS) 243 3.1.1. THE
SPLITTING METHOD BASED ON IMPLICIT SCHEMES OF THE FIRST ORDER OF
ACCURACY 243 3.1.2. THE METHOD OF COMPONENT SPLITTING BASED ON THE
CRANCK-NICHOLSON SCHEMES 243 3.2. METHODS OF TWO-CYCLIC MULTI-COMPONENT
SPLITTING 245 3.2.1. THE METHOD OF TWO-CYCLIC MULTI-COMPONENT SPLITTING
245 3.2.2. METHOD OF TWO-CYCLIC COMPONENT SPLITTING FOR QUASI-LINEAR
PROBLEMS .... 246 3.3. THE SPLITTING METHOD WITH FACTORISATION OF
OPERATORS 247 3.3.1. THE IMPLICIT SPLITTING SCHEME WITH APPROXIMATE
FACTORISATION OF THE OPERATOR 247 3.3.2. THE STABILISATION METHOD (THE
EXPLICIT-IMPLICIT SCHEMES WITH APPROXIMATE FACTORISATION OF THE
OPERATOR) 248 3.4. THE PREDICTOR-CORRECTOR METHOD 250 3.4.1. THE
PREDICTOR-CORRECTOR METHOD. THE CASE A =A T +A 2 250 3.4.2. THE
PREDICTOR-CORRECTOR METHOD. CASE A= _ 251 3.5. THE
ALTERNATING-DIRECTION METHOD AND THE METHOD OF THE STABILISING
CORRECTION 252 3.5.1. THE ALTERNATING-DIRECTION METHOD 252 3.5.2. THE
METHOD OF STABILISING CORRECTION 253 3.6. WEAK APPROXIMATION METHOD 254
3.6.1. THE MAIN SYSTEM OF PROBLEMS 254 3.6.2. TWO-CYCLIC METHOD OF WEAK
APPROXIMATION 254 3.7. THE SPLITTING METHODS - ITERATION METHODS OF
SOLVING STATIONARY PROBLEMS . 255 3.7.1. THE GENERAL CONCEPTS OF THE
THEORY OF ITERATION METHODS 255 3.7.2. ITERATION ALGORITHMS 256 4.
SPLITTING METHODS FOR APPLIED PROBLEMS OF MATHEMATICAL PHYSICS 257 4.1.
SPLITTING METHODS OF HEAT CONDUCTION EQUATIONS 258 4.1.1. THE FRACTIONAL
STEP METHOD 258 4.2.1. LOCALLY ONE-DIMENSIONAL SCHEMES. 259 4.1.3.
ALTERNATING-DIRECTION SCHEMES A 260 4.2. SPLITTING METHODS FOR
HYDRODYNAMICS PROBLEMS 262 4.2.1. SPLITTING METHODS FOR NAVIER-STOKES
EQUATIONS 262 4.2.2. THE FRACTIONAL STEPS METHOD FOR THE SHALLOW WATER
EQUATIONS 263 4.3. SPLITTING METHODS FOR THE MODEL OF DYNAMICS OF SEA
AND OCEAN FLOWS 268 4.3.1. THE NON-STATIONARY MODEL OF DYNAMICS OF SEA
AND OCEAN FLOWS 268 4.3.2. THE SPLITTING METHOD . 270 BIBLIOGRAPHIC
COMMENTARY. 272 7. METHODS FOR SOLVING NON-LINEAR EQUATIONS 273 MAIN
CONCEPTS AND DEFINITIONS 273 1. INTRODUCTION 274 2. ELEMENTS OF
NONLINEAR ANALYSIS 276 2.1. CONTINUITY AND DIFFERENTIABILITY OF
NONLINEAR MAPPINGS 276 2.1.1. MAIN DEFINITIONS 276 XIII 2.1.2.
DERIVATIVE AND GRADIENT OF THE FUNCTIONAL 277 2.1.3. DIFFERENTIABILITY
ACCORDING TO FRECHET 278 2.1.4. DERIVATIVES OF HIGH ORDERS AND TAYLOR
SERIES 278 2.2. ADJOINT NONLINEAR OPERATORS 279 2.2.1. ADJOINT NONLINEAR
OPERATORS AND THEIR PROPERTIES 279 2.2.2. SYMMETRY AND SKEW SYMMETRY
7... 280 2.3. CONVEX FUNCTIONALS AND MONOTONIC OPERATORS 280 2.4.
VARIATIONAL METHOD OF EXAMINING NONLINEAR EQUATIONS 282 2.4.1. EXTREME
AND CRITICAL POINTS OF FUNCTIONALS 282 2.4.2. THE THEOREMS OF EXISTENCE
OF CRITICAL POINTS 282 2.4.3. MAIN CONCEPT OF THE VARIATIONAL METHOD 283
2.4.4. THE SOLVABILITY OF THE EQUATIONS WITH MONOTONIC OPERATORS 283 2.5
MINIMISING SEQUENCES 284 2.5.1. MINIMIZING SEQUENCES AND THEIR
PROPERTIES 284 2.5.2. CORRECT FORMULATION OF THE MINIMISATION PROBLEM
285 3. THE METHOD OF THE STEEPEST DESCENT 285 3.1. NON-LINEAR EQUATION
AND ITS VARIATIONAL FORMULATION 285 3.2. MAIN CONCEPT OF THE STEEPEST
DESCENT METHODS 286 3.3. CONVERGENCE OF THE METHOD 287 4. THE RITZ
METHOD 288 4.1. APPROXIMATIONS AND RITZ SYSTEMS 289 4.2. SOLVABILITY OF
THE RITZ SYSTEMS 290 4.3. CONVERGENCE OF THE RITZ METHOD 291 5. THE
NEWTON-KANTOROVICH METHOD 291 5.1. DESCRIPTION OF THE NEWTON ITERATION
PROCESS 291 5.2. THE CONVERGENCE OF THE NEWTON ITERATION PROCESS 292
5.3. THE MODIFIED NEWTON METHOD 292 6. THE GALERKIN-PETROV METHOD FOR
NON-LINEAR EQUATIONS 293 6.1. APPROXIMATIONS AND GALERKIN SYSTEMS 293
6.2. RELATION TO PROJECTION METHODS 294 6.3. SOLVABILITY OF THE GALERKIN
SYSTEMS 295 6.4. THE CONVERGENCE OF THE GALERKIN-PETROV METHOD 295 7.
PERTURBATION METHOD 296 7.1. FORMULATION OF THE PERTURBATION ALGORITHM
296 7.2. JUSTIFICATION OF THE PERTURBATION ALGORITHMS 299 7.3. RELATION
TO THE METHOD OF SUCCESSIVE APPROXIMATIONS 301 8. APPLICATIONS TO SOME
PROBLEM OF MATHEMATICAL PHYSICS 302 8.1. THE PERTURBATION METHOD FOR A
QUASI-LINEAR PROBLEM OF NON-STATIONARY HEAT CONDUCTION 302 8.2. THE
GALERKIN METHOD FOR PROBLEMS OF DYNAMICS OF ATMOSPHERIC PROCESSES.. 306
8.3. THE NEWTON METHOD IN PROBLEMS OF VARIATIONAL DATA ASSIMILATION 308
BIBLIOGRAPHIC COMMENTARY 311 INDEX 317 XIV
|
| adam_txt |
METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS V.I. AGOSHKOV, P.B.
DUBOVSKI, V.P. SHUTYAEV CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING
CONTENTS PREFACE 1. MAIN PROBLEMS OF MATHEMATICAL PHYSICS 1 MAIN
CONCEPTS AND NOTATIONS 1 1. INTRODUCTION 2 2. CONCEPTS AND ASSUMPTIONS
FROM THE THEORY OF FUNCTIONS AND FUNCTIONAL ANALYSIS _ 3 2.1. POINT
SETS. CLASS OF FUNCTIONS C P (Q), C P (Q) 3 2.1.1. POINT SETS *. 3
2.1.2. CLASSES C(Q),(?(Q) 4 2.2. EXAMPLES FROM THE THEORY OF LINEAR
SPACES 5 2.2.1. NORMALISED SPACE 5 2.2.2. THE SPACE OF CONTINUOUS
FUNCTIONS C(Q) 6 2.2.3. SPACES C*(FI) 6 2.2.4. SPACE L P (Q) 7 2.3. L 2
(Q) SPACE. ORTHONORMAL SYSTEMS 9 2.3.1. " HILBERT SPACES 9 2.3.2. SPACE
L,(FI) 11 2.3.3. ORTHONORMAL SYSTEMS 11 2.4. LINEAR OPERATORS AND
FUNCTIONAL 13 2.4.1. LINEAR OPERATORS AND FUNCTIONALS 13 2.4.2. INVERSE
OPERATORS 15 2.4.3. ADJOINT, SYMMETRIC AND SELF-ADJOINT OPERATORS 15
2.4.4. POSITIVE OPERATORS AND ENERGETIC SPACE 16 2.4.5. LINEAR
EQUATIONS ;. 17 2.4.6. EIGENVALUE PROBLEMS .' 17 2.5. GENERALIZED
DERIVATIVES. SOBOLEV SPACES 19 2.5.1. GENERALIZED DERIVATIVES 19 2.5.2.
SOBOLEV SPACES 20 2.5.3. THE GREEN FORMULA '. 21 3. MAIN EQUATIONS AND
PROBLEMS OF MATHEMATICAL PHYSICS 22 3.1. MAIN EQUATIONS OF MATHEMATICAL
PHYSICS 22 3.1.1. LAPLACE AND POISSON EQUATIONS 23 .2. EQUATIONS OF
OSCILLATIONS 24 .3. HELMHOLTZ EQUATION 26 .4. DIFFUSION AND HEAT
CONDUCTION EQUATIONS 26 .5. MAXWELL AND TELEGRAPH EQUATIONS 27 .6.
TRANSFER EQUATION 28 .7. GAS- AND HYDRODYNAMIC EQUATIONS : 29 .8.
CLASSIFICATION OF LINEAR DIFFERENTIAL EQUATIONS 29 VII 3.2. FORMULATION
OF THE MAIN PROBLEMS OF MATHEMATICAL PHYSICS 32 3.2.1. CLASSIFICATION OF
BOUNDARY-VALUE PROBLEMS 32 3.2.2. THE CAUCHY PROBLEM 33 3.2.3. THE
BOUNDARY-VALUE PROBLEM FOR THE ELLIPTICAL EQUATION 34 3.2.4. MIXED
PROBLEMS 35 3.2.5. VALIDITY OF FORMULATION OF PROBLEMS.
CAUCHY-KOVALEVSKII THEOREM 35 3.3. GENERALIZED FORMULATIONS AND
SOLUTIONS OF MATHEMATICAL PHYSICS PROBLEMS. 37 3.3.1. GENERALIZED
FORMULATIONS AND SOLUTIONS OF ELLIPTICAL PROBLEMS 38 3.3.2. GENERALIZED
FORMULATIONS AND SOLUTION OF HYPERBOLIC PROBLEMS 41 3.3.3. THE
GENERALIZED FORMULATION AND SOLUTIONS OF PARABOLIC PROBLEMS 43 3.4.
VARIATIONAL FORMULATIONS OF PROBLEMS 45 3.4.1. VARIATIONAL FORMULATION
OF PROBLEMS IN THE CASE OF POSITIVE DEFINITE OPERATORS 45 3.4.2.
VARIATIONAL FORMULATION OF THE PROBLEM IN THE CASE OF POSITIVE OPERATORS
. 46 3.4.3. VARIATIONAL FORMULATION OF THE BASIC ELLIPTICAL PROBLEMS 47
3.5. INTEGRAL EQUATIONS 49 3.5.1. INTEGRAL FREDHOLM EQUATION OF THE 1ST
AND 2ND KIND 49 3.5.2. VOLTERRA INTEGRAL EQUATIONS 50 3.5.3. INTEGRAL
EQUATIONS WITH A POLAR KERNEL 51 3.5.4. FREDHOLM THEOREM 51 3.5.5.
INTEGRAL EQUATION WITH THE HERMITIAN KERNEL 52 BIBLIOGRAPHIC COMMENTARY
54 2. METHODS OF POTENTIAL THEORY 56 MAIN CONCEPTS AND DESIGNATIONS H 56
1. INTRODUCTION 57 2. FUNDAMENTALS OF POTENTIAL THEORY 58 2.1.
ADDITIONAL INFORMATION FROM MATHEMATICAL ANALYSIS 58 2.1.1 MAIN
ORTHOGONAL COORDINATES 58 2.1.2. MAIN DIFFERENTIAL OPERATIONS OF THE
VECTOR FIELD 58 2.1.3. FORMULAE FROM THE FIELD THEORY 59 2.1.4. MAIN
PROPERTIES OF HARMONIC FUNCTIONS 69 2.2 POTENTIAL OF VOLUME MASSES OR
CHARGES 61 2.2.1. NEWTON (COULOMB) POTENTIAL 61 2.2.2. THE PROPERTIES OF
THE NEWTON POTENTIAL 61 2.2.3. POTENTIAL OF A HOMOGENEOUS SPHERE 62
2.2.4. PROPERTIES OF THE POTENTIAL OF VOLUME-DISTRIBUTED MASSES 62 2.3.
LOGARITHMIC POTENTIAL 63 2.3.1. DEFINITION OF THE LOGARITHMIC POTENTIAL
63 2.3.2. THE PROPERTIES OF THE LOGARITHMIC POTENTIAL 63 2.3.3. THE
LOGARITHMIC POTENTIAL OF A CIRCLE WITH CONSTANT DENSITY 64 2.4. THE
SIMPLE LAYER POTENTIAL 64 2.4.1. DEFINITION OF THE SIMPLE LAYER
POTENTIAL IN SPACE 64 2.4.2. THE PROPERTIES OF THE SIMPLE LAYER
POTENTIAL 65 2.4.3. THE POTENTIAL OF THE HOMOGENEOUS SPHERE 66 2.4.4.
THE SIMPLE LAYER POTENTIAL ON A PLANE 66 VNI 2.5. DOUBLE LAYER POTENTIAL
67 2.5.1. DIPOLE POTENTIAL 67 2.5.2. THE DOUBLE LAYER POTENTIAL IN SPACE
AND ITS PROPERTIES 67 2.5.3. THE LOGARITHMIC DOUBLE LAYER POTENTIAL AND
ITS PROPERTIES 69 3. USING THE POTENTIAL THEORY IN CLASSIC PROBLEMS OF
MATHEMATICAL PHYSICS 70 3.1. SOLUTION OF THE LAPLACE AND POISSON
EQUATIONS 70 3.1.1. FORMULATION OF THE BOUNDARY-VALUE PROBLEMS OF THE
LAPLACE EQUATION .70 3.1.2 SOLUTION OF THE DIRICHLET PROBLEM IN SPACE
71 3.1.3. SOLUTION OF THE DIRICHLET PROBLEM ON A PLANE 72 3.1.4.
SOLUTION OF THE NEUMANN PROBLEM 73 3.1.5. SOLUTION OF THE THIRD
BOUNDARY-VALUE PROBLEM FOR THE LAPLACE EQUATION .74 3.1.6. SOLUTION OF
THE BOUNDARY-VALUE PROBLEM FOR THE POISSON EQUATION 75 3.2. THE GREEN
FUNCTION OF THE LAPLACE OPERATOR 76 3.2.1. THE POISSON EQUATION 76
3.2.2. THE GREEN FUNCTION 76 3.2.3. SOLUTION OF THE DIRICHLET PROBLEM
FOR SIMPLE DOMAINS 77 3.3 SOLUTION OF THE LAPLACE EQUATION FOR COMPLEX
DOMAINS 78 3.3.1. SCHWARZ METHOD 78 3.3.2. THE SWEEP METHOD 80 4. OTHER
APPLICATIONS OF THE POTENTIAL METHOD 81 4.1. APPLICATION OF THE
POTENTIAL METHODS TO THE HELMHOLTZ EQUATION 81 4.1.1. MAIN FACTS ,. 81
4.1.2. BOUNDARY-VALUE PROBLEMS FOR THE HELMHOLTZ EQUATIONS 82 4.1.3.
GREEN FUNCTION 84 4.1.4. EQUATION AV-XV - 0 85 4.2. NON-STATIONARY
POTENTIALS ; 86 4.2.1 POTENTIALS FOR THE ONE-DIMENSIONAL HEAT EQUATION
86 4.2.2. HEAT SOURCES IN MULTIDIMENSIONAL CASE 88 4.2.3. THE
BOUNDARY-VALUE PROBLEM FOR THE WAVE EQUATION 90 BIBLIOGRAPHIC COMMENTARY
92, 3. EIGENFUNCTION METHODS 1 94 MAIN CONCEPTS AND NOTATIONS .: 94 1.
INTRODUCTION 94 2. EIGENVALUE PROBLEMS 95 2.1. FORMULATION AND THEORY 95
2.2. EIGENVALUE PROBLEMS FOR DIFFERENTIAL OPERATORS 98 2.3. PROPERTIES
OF EIGENVALUES AND EIGENFUNCTIONS 99 2.4. FOURIER SERIES 100 2.5.
EIGENFUNCTIONS OF SOME ONE-DIMENSIONAL PROBLEMS 102 3. SPECIAL FUNCTIONS
103 3.1. SPHERICAL FUNCTIONS 103 3.2. LEGENDRE POLYNOMIALS 105 3.3.
CYLINDRICAL FUNCTIONS 106 3.4. CHEBYSHEF, LAGUERRE AND HERMITE
POLYNOMIALS 107 3.5. MATHIEU FUNCTIONS AND HYPERGEOMETRICAL FUNCTIONS
109 4. EIGENFUNCTION METHOD 110 4.1. GENERAL SCHEME OF THE EIGENFUNCTION
METHOD .* 110 4.2. THE EIGENFUNCTION METHOD FOR DIFFERENTIAL EQUATIONS
OF MATHEMATICAL PHYSICS ILL 4.3. SOLUTION OF PROBLEMS WITH
NONHOMOGENEOUS BOUNDARY CONDITIONS 114 5. EIGENFUNCTION METHOD FOR
PROBLEMS OF THE THEORY OF ELECTROMAGNETIC PHENOMENA 115 5.1. THE PROBLEM
OF ABOUNDED TELEGRAPH LINE 115 5.2. ELECTROSTATIC FIELD INSIDE AN
INFINITE PRISM 117 5.3. PROBLEM OF THE ELECTROSTATIC FIELD INSIDE A
CYLINDER 117 5.4. THE FIELD INSIDE A BALL AT A GIVEN POTENTIAL ON ITS
SURFACE 118 5.5 THE FIELD OF A CHARGE INDUCED ON A BALL 120. 6.
EIGENFUNCTION METHOD FOR HEAT CONDUCTIVITY PROBLEMS 121 6.1. HEAT
CONDUCTIVITY IN A BOUNDED BAR 121 6.2. STATIONARY DISTRIBUTION OF
TEMPERATURE IN AN INFINITE PRISM 122 6.3. TEMPERATURE DISTRIBUTION OF A
HOMOGENEOUS CYLINDER 123 7. EIGENFUNCTION METHOD FOR PROBLEMS IN THE
THEORY OF OSCILLATIONS 124 7.1. FREE OSCILLATIONS OF A HOMOGENEOUS
STRING 124 7.2. OSCILLATIONS OF THE STRING WITH A MOVING END 125 7.3.
PROBLEM OF ACOUSTICS OF FREE OSCILLATIONS OF GAS 126 7.4. OSCILLATIONS
OF A MEMBRANE WITH A FIXED END 127 7.5. PROBLEM OF OSCILLATION OF A
CIRCULAR MEMBRANE 128 BIBLIOGRAPHIC COMMENTARY 129 4. METHODS OFJNTEGRAL
TRANSFORMS 130 MAIN CONCEPTS AND DEFINITIONS 130 1. INTRODUCTION 131 2.
MAIN INTEGRAL TRANSFORMATIONS'.: 132 2.1. FOURIER TRANSFORM 132 2.1.1.
THE MAIN PROPERTIES OF FOURIER TRANSFORMS 133 2.1.2. MULTIPLE FOURIER
TRANSFORM 134 2.2. LAPLACE TRANSFORM !*. 134 2.2.1. LAPLACE INTEGRAL .'
134 2.2.2. THE INVERSION FORMULA FOR THE LAPLACE TRANSFORM 135 2.2.3.
MAIN FORMULAE AND LIMITING THEOREMS 135 2.3. MELLIN TRANSFORM 135 2.4.
HANKEL TRANSFORM 136 2.5. MEYER TRANSFORM 138 2.6. KONTOROVICH-LEBEDEV
TRANSFORM 138 2.7. MELLER-FOCK TRANSFORM 139 2.8 HILBERT TRANSFORM 140
2.9. LAGUERRE AND LEGENDRE TRANSFORMS 140 2.10 BOCHNER AND CONVOLUTION
TRANSFORMS, WAVELETS AND CHAIN TRANSFORMS 141 3. USING INTEGRAL
TRANSFORMS IN PROBLEMS OF OSCILLATION THEORY 143 3.1. ELECTRICAL
OSCILLATIONS 143 3.2. TRANSVERSE VIBRATIONS OF A STRING 143 3.3.
TRANSVERSE VIBRATIONS OF AN INFINITE CIRCULAR MEMBRANE 146 4. USING
INTEGRAL TRANSFORMS IN HEAT CONDUCTIVITY PROBLEMS 147 4.1. SOLVING HEAT
CONDUCTIVITY PROBLEMS USING THE LAPLACE TRANSFORM 147 4.2. SOLUTION OF A
HEAT CONDUCTIVITY PROBLEM USING FOURIER TRANSFORMS 148 4.3. TEMPERATURE
REGIME OF A SPHERICAL BALL 149 5. USING INTEGRAL TRANSFORMATIONS IN THE
THEORY OF NEUTRON DIFFUSION 149 5.1. THE SOLUTION OF THE EQUATION OF
DECELERATION OF NEUTRONS FOR A MODERATOR OF INFINITE DIMENSIONS 150 5.2.
THE PROBLEM OF DIFFUSION OF THERMAL NEUTRONS 150 6. APPLICATION OF
INTEGRAL TRANSFORMATIONS TO HYDRODYNAMIC PROBLEMS 151 6.1. A
TWO-DIMENSIONAL VORTEX-FREE FLOW OF AN IDEAL LIQUID 151 6.2. THE FLOW OF
THE IDEAL LIQUID THROUGH A SLIT 152 6.3. DISCHARGE OF THE IDEAL LIQUID
THROUGH A CIRCULAR ORIFICE 153 7. USING INTEGRAL TRANSFORMS IN
ELASTICITY THEORY 155 7.1. AXISYMMETRIC STRESSES IN ACYLINDER 155 7.2.
BUSSINESQ PROBLEM;FOR THE HALF SPACE 157 7.3. DETERMINATION OF STRESSES
IN A WEDGE 158 8. USING INTEGRAL TRANSFORMS IN COAGULATION KINETICS 159
8.1. EXACT SOLUTION OF THE COAGULATION EQUATION 159 8.2. VIOLATION OF
THE MASS CONSERVATION LAW 161 BIBLIOGRAPHIC COMMENTARY .; 162 5. METHODS
OF DISCRETISATION OF MATHEMATICAL PHYSICS PROBLEMS 163 MAIN DEFINITIONS
AND NOTATIONS 163 1. INTRODUCTION '. 164 2. FINITE-DIFFERENCE METHODS
166 2.1. THE NET METHOD 166 2.1.1. MAIN CONCEPTS AND DEFINITIONS OF THE
METHOD 166 2.1.2. GENERAL DEFINITIONS OF THE NET METHOD. THE CONVERGENCE
THEOREM 170, 2.1.3. THE NET METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
173 2.2. THE METHOD OF ARBITRARY LINES :. 182 2.2.1. THE METHOD OF
ARBITRARY LINES FOR PARABOLIC-TYPE EQUATIONS 182 2.2.2. THE METHOD OF
ARBITRARY LINES FOR HYPERBOLIC EQUATIONS 184 2.2.3. THE METHOD OF
ARBITRARY LINES FOR ELLIPTICAL EQUATIONS 185 2.3. THE NET METHOD FOR
INTEGRAL EQUATIONS (THE QUADRATURE METHOD) 187 3. VARIATIONAL METHODS
188 3.1. MAIN CONCEPTS OF VARIATIONAL FORMULATIONS OF PROBLEMS AND
VARIATIONAL METHODS .188 3.1.1. VARIATIONAL FORMULATIONS OF PROBLEMS
188 3.1.2. CONCEPTS OF THE DIRECT METHODS IN CALCULUS OF VARIATIONS 189
3.2. THE RITZ METHOD 190 3.2.1. THE CLASSIC RITZ METHOD 190 3.2.2. THE
RITZ METHOD IN ENERGY SPACES 192 3.2.3. NATURAL AND MAIN BOUNDARY-VALUE
CONDITIONS 194 3.3. THE METHOD OF LEAST SQUARES 195 XI 3.4. KANTOROVICH,
COURANT AND TREFFTZ METHODS 196 3.4.1. THE KANTOROVICH METHOD 1% 3.4.2.
COURANT METHOD 1% 3.4.3. TREFFTZ METHOD , 197 3.5. VARIATIONAL METHODS
IN THE EIGENVALUE PROBLEM 199 4. PROJECTION METHODS 201 4.1.
THEBUBNOV-GALERKIN METHOD V 201 4.1.1. THEBUBNOV-GALERKIN METHOD (A
GENERAL CASE) 201 4.1.2 THEBUBNOV-GALERKIN METHOD (I4= \ 0 +J5) 202 4.2.
THE MOMENTS METHOD 204 4.3. PROJECTION METHODS IN THE HILBERT AND BANACH
SPACES 205 4.3.1. THE PROJECTION METHOD IN THE HILBERT SPACE 205 4.3.2.
THE GALERKIN-PETROV METHOD 206 4.3.3. THE PROJECTION METHOD IN THE
BANACH SPACE 206 4.3.4. THE COLLOCATION METHOD 208 4.4. MAIN CONCEPTS OF
THE PROJECTION-GRID METHODS 208 5. METHODS OF INTEGRAL IDENTITIES 210
5.1. THE MAIN CONCEPTS OF THE METHOD 210 5.2. THE METHOD OF MARCHUK'S
INTEGRAL IDENTITY 211 5.3. GENERALIZED FORMULATION OF THE METHOD OF
INTEGRAL IDENTITIES 213 5.3.1. ALGORITHM OF CONSTRUCTING INTEGRAL
IDENTITIES 213 5.3.2. THE DIFFERENCE METHOD OF APPROXIMATING THE
INTEGRAL IDENTITIES 214 5.3.3. THE PROJECTION METHOD OF APPROXIMATING
THE INTEGRAL IDENTITIES 215 5.4. APPLICATIONS OF THE METHODS OF INTEGRAL
IDENTITIES IN MATHEMATICAL PHYSICS PROBLEMS 217 5.4.1. THE METHOD OF
INTEGRAL IDENTITIES FOR THE DIFFUSION EQUATION 217 5.4.2. THE SOLUTION
OF DEGENERATING EQUATIONS 219 5.4.3. THE METHOD OF INTEGRAL IDENTITIES
FOR EIGENVALUE PROBLEMS 221 BIBLIOGRAPHIC COMMENTARY 223 6. SPLITTING
METHODS 224 1. INTRODUCTION 224 2. INFORMATION FROM THE THEORY OF
EVOLUTION EQUATIONS AND DIFFERENCE SCHEMES . 225 2.1. EVOLUTION
EQUATIONS 225 2.1.1. THE CAUCHY PROBLEM 225 2.1.2. THE NONHOMOGENEOUS
EVOLUTION EQUATION 228 2.1.3. EVOLUTION EQUATIONS WITH BOUNDED OPERATORS
229 2.2. OPERATOR EQUATIONS IN FINITE-DIMENSIONAL SPACES 231 2.2.1. THE
EVOLUTION SYSTEM 231 2.2.2. STATIONARISATION METHOD 232 2.3. CONCEPTS
AND INFORMATION FROM THE THEORY OF DIFFERENCE SCHEMES 233 2.3.1.
APPROXIMATION 233 2.3.2. STABILITY 239 2.3.3. CONVERGENCE : 240 2.3.4.
THE SWEEP METHOD 241 3. SPLITTING METHODS .: 242 XII 3.1. THE METHOD OF
COMPONENT SPLITTING (THE FRACTIONAL STEP METHODS) 243 3.1.1. THE
SPLITTING METHOD BASED ON IMPLICIT SCHEMES OF THE FIRST ORDER OF
ACCURACY 243 3.1.2. THE METHOD OF COMPONENT SPLITTING BASED ON THE
CRANCK-NICHOLSON SCHEMES 243 3.2. METHODS OF TWO-CYCLIC MULTI-COMPONENT
"SPLITTING 245 3.2.1. THE METHOD OF TWO-CYCLIC MULTI-COMPONENT SPLITTING
245 3.2.2. METHOD OF TWO-CYCLIC COMPONENT SPLITTING FOR QUASI-LINEAR
PROBLEMS . 246 3.3. THE SPLITTING METHOD WITH FACTORISATION OF
OPERATORS 247 3.3.1. THE IMPLICIT SPLITTING SCHEME WITH APPROXIMATE
FACTORISATION OF THE OPERATOR 247 3.3.2. THE STABILISATION METHOD (THE
EXPLICIT-IMPLICIT SCHEMES WITH APPROXIMATE FACTORISATION OF THE
OPERATOR) 248 3.4. THE PREDICTOR-CORRECTOR METHOD 250 3.4.1. THE
PREDICTOR-CORRECTOR METHOD. THE CASE A =A T +A 2 250 3.4.2. THE
PREDICTOR-CORRECTOR METHOD. CASE A= _ \ 251 3.5. THE
ALTERNATING-DIRECTION METHOD AND THE METHOD OF THE STABILISING
CORRECTION 252 3.5.1. THE ALTERNATING-DIRECTION METHOD 252 3.5.2. THE
METHOD OF STABILISING CORRECTION 253 3.6. WEAK APPROXIMATION METHOD 254
3.6.1. THE MAIN SYSTEM OF PROBLEMS 254 3.6.2. TWO-CYCLIC METHOD OF WEAK
APPROXIMATION 254 3.7. THE SPLITTING METHODS - ITERATION METHODS OF
SOLVING STATIONARY PROBLEMS . 255 3.7.1. THE GENERAL CONCEPTS OF THE
THEORY OF ITERATION METHODS 255 3.7.2. ITERATION ALGORITHMS 256 4.
SPLITTING METHODS FOR APPLIED PROBLEMS OF MATHEMATICAL PHYSICS 257 4.1.
SPLITTING METHODS OF HEAT CONDUCTION EQUATIONS 258 4.1.1. THE FRACTIONAL
STEP METHOD 258 4.2.1. LOCALLY ONE-DIMENSIONAL SCHEMES. 259 4.1.3.
ALTERNATING-DIRECTION SCHEMES A 260 4.2. SPLITTING METHODS FOR
HYDRODYNAMICS PROBLEMS 262 4.2.1. SPLITTING METHODS FOR NAVIER-STOKES
EQUATIONS 262 4.2.2. THE FRACTIONAL STEPS METHOD FOR THE SHALLOW WATER
EQUATIONS 263 4.3. SPLITTING METHODS FOR THE MODEL OF DYNAMICS OF SEA
AND OCEAN FLOWS 268 4.3.1. THE NON-STATIONARY MODEL OF DYNAMICS OF SEA
AND OCEAN FLOWS 268 4.3.2. THE SPLITTING METHOD '. 270 BIBLIOGRAPHIC
COMMENTARY. 272 7. METHODS FOR SOLVING NON-LINEAR EQUATIONS 273 MAIN
CONCEPTS AND DEFINITIONS 273 1. INTRODUCTION 274 2. ELEMENTS OF
NONLINEAR ANALYSIS 276 2.1. CONTINUITY AND DIFFERENTIABILITY OF
NONLINEAR MAPPINGS 276 2.1.1. MAIN DEFINITIONS 276 XIII 2.1.2.
DERIVATIVE AND GRADIENT OF THE FUNCTIONAL 277 2.1.3. DIFFERENTIABILITY
ACCORDING TO FRECHET 278 2.1.4. DERIVATIVES OF HIGH ORDERS AND TAYLOR
SERIES 278 2.2. ADJOINT NONLINEAR OPERATORS 279 2.2.1. ADJOINT NONLINEAR
OPERATORS AND THEIR PROPERTIES 279 2.2.2. SYMMETRY AND SKEW SYMMETRY
7. 280 2.3. CONVEX FUNCTIONALS AND MONOTONIC OPERATORS 280 2.4.
VARIATIONAL METHOD OF EXAMINING NONLINEAR EQUATIONS 282 2.4.1. EXTREME
AND CRITICAL POINTS OF FUNCTIONALS 282 2.4.2. THE THEOREMS OF EXISTENCE
OF CRITICAL POINTS 282 2.4.3. MAIN CONCEPT OF THE VARIATIONAL METHOD 283
2.4.4. THE SOLVABILITY OF THE EQUATIONS WITH MONOTONIC OPERATORS 283 2.5
MINIMISING SEQUENCES 284 2.5.1. MINIMIZING SEQUENCES AND THEIR
PROPERTIES 284 2.5.2. CORRECT FORMULATION OF THE MINIMISATION PROBLEM
285 3. THE METHOD OF THE STEEPEST DESCENT 285 3.1. NON-LINEAR EQUATION
AND ITS VARIATIONAL FORMULATION 285 3.2. MAIN CONCEPT OF THE STEEPEST
DESCENT METHODS 286 3.3. CONVERGENCE OF THE METHOD 287 4. THE RITZ
METHOD 288 4.1. APPROXIMATIONS AND RITZ SYSTEMS 289 4.2. SOLVABILITY OF
THE RITZ SYSTEMS 290 4.3. CONVERGENCE OF THE RITZ METHOD 291 5. THE
NEWTON-KANTOROVICH METHOD 291 5.1. DESCRIPTION OF THE NEWTON ITERATION
PROCESS 291 5.2. THE CONVERGENCE OF THE NEWTON ITERATION PROCESS 292
5.3. THE MODIFIED NEWTON METHOD 292 6. THE GALERKIN-PETROV METHOD FOR
NON-LINEAR EQUATIONS 293 6.1. APPROXIMATIONS AND GALERKIN SYSTEMS 293
6.2. RELATION TO PROJECTION METHODS 294 6.3. SOLVABILITY OF THE GALERKIN
SYSTEMS 295 6.4. THE CONVERGENCE OF THE GALERKIN-PETROV METHOD 295 7.
PERTURBATION METHOD 296 7.1. FORMULATION OF THE PERTURBATION ALGORITHM
296 7.2. JUSTIFICATION OF THE PERTURBATION ALGORITHMS 299 7.3. RELATION
TO THE METHOD OF SUCCESSIVE APPROXIMATIONS 301 8. APPLICATIONS TO SOME
PROBLEM OF MATHEMATICAL PHYSICS 302 8.1. THE PERTURBATION METHOD FOR A
QUASI-LINEAR PROBLEM OF NON-STATIONARY HEAT CONDUCTION 302 8.2. THE
GALERKIN METHOD FOR PROBLEMS OF DYNAMICS OF ATMOSPHERIC PROCESSES. 306
8.3. THE NEWTON METHOD IN PROBLEMS OF VARIATIONAL DATA ASSIMILATION 308
BIBLIOGRAPHIC COMMENTARY 311 INDEX 317 XIV |
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| author | Agoškov, Vasilij I. Dubovski, Pavel B. Šutjaev, Viktor P. |
| author_facet | Agoškov, Vasilij I. Dubovski, Pavel B. Šutjaev, Viktor P. |
| author_role | aut aut aut |
| author_sort | Agoškov, Vasilij I. |
| author_variant | v i a vi via p b d pb pbd v p š vp vpš |
| building | Verbundindex |
| bvnumber | BV023253394 |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)249527235 (DE-599)BSZ259142743 |
| 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 | 1. publ. |
| format | Book |
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| id | DE-604.BV023253394 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:28:51Z |
| indexdate | 2024-07-09T21:14:11Z |
| institution | BVB |
| isbn | 9781904602057 1904602053 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016438722 |
| oclc_num | 249527235 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XIV, 320 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Cambridge Internat. Science Publ. |
| record_format | marc |
| spelling | Agoškov, Vasilij I. Verfasser aut Methods for solving mathematical physics problems V. I. Agoshkov ; P. B. Dubovski ; V. P. Shutyaev 1. publ. Cambridge Cambridge Internat. Science Publ. 2006 XIV, 320 S. txt rdacontent n rdamedia nc rdacarrier Mathematical physics - Methodology Mathematische Physik - Mathematische Methode Mathematische Physik Mathematical physics Methodology Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s Mathematische Methode (DE-588)4155620-3 s DE-604 Dubovski, Pavel B. Verfasser aut Šutjaev, Viktor P. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016438722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Agoškov, Vasilij I. Dubovski, Pavel B. Šutjaev, Viktor P. Methods for solving mathematical physics problems Mathematical physics - Methodology Mathematische Physik - Mathematische Methode Mathematische Physik Mathematical physics Methodology Mathematische Methode (DE-588)4155620-3 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4155620-3 (DE-588)4037952-8 |
| title | Methods for solving mathematical physics problems |
| title_auth | Methods for solving mathematical physics problems |
| title_exact_search | Methods for solving mathematical physics problems |
| title_exact_search_txtP | Methods for solving mathematical physics problems |
| title_full | Methods for solving mathematical physics problems V. I. Agoshkov ; P. B. Dubovski ; V. P. Shutyaev |
| title_fullStr | Methods for solving mathematical physics problems V. I. Agoshkov ; P. B. Dubovski ; V. P. Shutyaev |
| title_full_unstemmed | Methods for solving mathematical physics problems V. I. Agoshkov ; P. B. Dubovski ; V. P. Shutyaev |
| title_short | Methods for solving mathematical physics problems |
| title_sort | methods for solving mathematical physics problems |
| topic | Mathematical physics - Methodology Mathematische Physik - Mathematische Methode Mathematische Physik Mathematical physics Methodology Mathematische Methode (DE-588)4155620-3 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematical physics - Methodology Mathematische Physik - Mathematische Methode Mathematische Physik Mathematical physics Methodology Mathematische Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016438722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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