Some classes of partial differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Gordon and Breach
1988
|
| Series: | Advanced studies in contemporary mathematics
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XI, 504 S. |
| ISBN: | 2881246621 |
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MARC
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| 100 | 1 | |a Bicadze, Andrej V. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Nekotorye klassy uravnenij v častnych proizvodnych |
| 245 | 1 | 0 | |a Some classes of partial differential equations |c A. V. Bitsadze |
| 264 | 1 | |a New York [u.a.] |b Gordon and Breach |c 1988 | |
| 300 | |a XI, 504 S. | ||
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| 490 | 1 | |a Advanced studies in contemporary mathematics |v 4 | |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Preface xiii
Chapter I. Introduction 1
§1. The Classification of partial differential equations 1
1 The concept of the partial differential equations 1
2 Division by type of partial differential equation 2
3 Linear second order partial differential equations 4
4 Linear systems of second order partial differential
equations 10
5 The characteristic curves and characteristic surfaces of
linear second order equations 12
§2. Extremum principles 16
1 The concepts of classes of smoothness of functions and the
smoothness of the boundary of the domain of their
specification 16
2 The extremum principle for solutions of second order
elliptic equations 18
3 The extremum principle for solutions of the heat equation 22
4 The case of hyperbolic equations 24
§3. Modelling some physical phenomena in terms of partial
differential equations 27
1 Oscillations of elastic material continua 27
2 Heat propagation 31
3 Plane parallel flow of a compressible medium 33
4 Chaplygin equation 37
vi A. V. BITSADZE
5 Approximate replacement of transonic flow by hodograph
plane 40
§4. General remarks with respect to the structural properties of
solutions of partial differential equations and the formulation
of linear problems for these equations 45
1 Directing arguments 45
2 Prospects for constructing classes of solutions of partial
differential equations 50
3 Second order hyperbolic equations 52
4 The structural properties of solutions of one class of
second order hyperbolic systems with two independent
variables 61
5 Isolated singular points of the harmonic function 67
6 The integral representation of harmonic functions and
some of its applications 73
7 Singularities of solutions of wave equations 83
8 Integral representation of solutions of a wave equation with
two spatial variables 86
9 Heat equation 88
§5. Brief review of the known methods of solving partial
differential equations 90
1 The general formulation of boundary value problems for
second order elliptic equations 90
2 The Parametrix method 92
3 An elementary solution of second order elliptic equations 95
4 Integral representation of solutions of second order elliptic
equations 99
5 The potential method 101
6 The method of integral equations 104
7 Linear equations in linear normalized spaces 111
8 Conditional classification of problems for partial
differential equations 116
§6. Generalized solutions of classical problems 118
CONTENTS vii
1 The degree of smoothness of solutions of variational
problems and a generalization of the concept of the
derivative 118
2 The spaces Wg and Wa 120
3 Generalized solution of the Dirichlet s problems for a
second order uniformly elliptic equation 122
4 Generalized solution of an inhomogeneous Dirichlet
problem (1.280), (1.281) 125
5 Generalization of the concept of the solution of a basic
compound problem for a second order hyperbolic equation 128
6 Generalized solutions of the first boundary value problem
for a second order equation 130
§7. Review of the methods of proving the existence of generalized
solutions 132
1 General Remarks 132
2 The orthogonal projection method 133
3 The existence of a generalized solution of a basic
compound problem for an inhomogenous wave equation 135
4 Existence of a generalized solution of the first boundary
value problem for an inhomogeneous heat equation 139
Chapter II. Second order elliptic equations 142
§ 1. Uniformly elliptic systems with split principal parts 142
1 General remarks 142
2 General complex representation of analytic solutions of
system (2.8) 146
3 Integral representations of analytic functions of one
complex variable 154
4 The Dirichlet problem for system (2.28) 157
5 Dirichlet problem for the elliptic system (2.8) 159
6 Poincare s problem for system (2.8) 165
7 Some special cases of problem (2.8), (2.64) 170
§2. Uniformly elliptic systems with non split principal parts 172
1 Integral representation of solutions of a uniform elliptic
system using the matrix parametrix method 172
viii A. V. BITSADZE
2 General representation of solutions of an elliptic system
with constant coefficients 179
3 The Dirichlet problem for strongly connected systems 184
4 A class of elliptic systems and domains for which
Dirichlet s problem is normally Hausdorff solvable 187
5 Elliptic systems in multi dimensional domains and classes
of systems with a different character of ellipticity in the
domain of their specification 194
§3. Second order elliptic equations with parabolic degeneracy on
the boundary of the domain of their specification L96
1 Preliminary remarks 196
2 Dirichlet s problem for Eqs.(2.141),(2.142),(2.143) in a •
domain lying in the upper half plane together with the
boundary 202
3 The case of Eq. (2.144) 205
4 The case of Eq. (2.145) 208
5 Some generalizations of the results of previous sections 211
6 The special case of Eq. (2.143) 215
§4. Some other problems for elliptic equations 219
1 The problem with a directional derivative 219
2 The directional derivative problem with polynomial
coefficients for harmonic functions 226
3 Some simpler generalizations of linear elliptic boundary
value problems 235
4 The linearized Navier S tokes problem 241
5 Final remarks 246
Chapter in. Second order hyperbolic equations 251
§1. Second order hyperbolic equations with split principal parts
when there is no parabolic degeneracy 251
1 Darboux s problem for the string oscillation equation 251
2 Darboux s problem for system (1.156) 255
3 Multi dimensional analogs of Darboux s problem 257
4 Some other versions of the characteristic Cauchy problem
and the Darboux problem 258
CONTENTS ix
§2. Second order hyperbolic systems with non split principal parts
when there is no hyperbolic degeneracy 264
1 Difficulties in formulating a characteristic problem for
hyperbolic systems 264
2 The effect of the influence of lowest terms 270
3 Normally hyperbolic systems and the formulation of a
characteristic problem for them 275
4 Reduction of problem (3.71),(3.72), (3.74) to a functional
equation 279
5 Analysis of the functional equation (3.83) 282
6 The hyperbolic system (3.74) when the condition of normal
hyperbolicity is violated 286
§3 Second order hyperbolic equations with parabolic degeneracy 288
1 Cauchy s problem with initial data on a line of parabolic
degeneration which is a set of cuspidal points of families of
characteristics 288
2 The case of Eq (3.95) without lowest terms 290
3 ThecaseofEq.(3.95) 293
4 The case of a degenerating hyperbolic system 296
5 A homogeneous equation corresponding to Eq. (1.33) 297
§4. Hyperbolic equations with degeneracy of type and order 302
1 The case of two independent variables 302
2 Eq. (2.176) when a alters outside the half open interval
li m, 1) 305
3 Darboux s first problem for Eq. (2.176) 310
4 The modified Darboux first problem 312
5 Multi dimensional analog of problem (1.54), (3.172) 317
Chapter IV. Equations of mixed type 323
§l.Tricomi s problem 323
1 History of the problem of equations of mixed type 323
2 The formulation of Tricomi s problem for Eq (2.176) 325
3 The extremum principle and the uniqueness of the solution
of problem T 327
x A. V. BITSADZE
4 Reduction of problem T to singular integral equations 329
5 The proof of the existence of solutions of singular integral
equations obtained in the previous section 342
6 Other methods of solving problem T for Eq. (L) 347
§2. Some examples and simple generalizations 358
1 Problem T in the case when D+ is a semi strip 358
2 Problem Ti
3 Problem T2 363
4 Problem T3 368
5 Frankl s problem 371
6 The case of Eq. (4.116) 377
§ 3. The general mixed problem 379
1 Formulation of the general mixed problem and its
correctness 379
2 Problem M 388
3 The existence of a solution of problem M 392
4 The solution of problem (4.199) when a = a0, v = 0 and
X = const 400
5 Some remarks on the general mixed problem 406
§4. Equations of mixed type in multi dimensional domains and
some problems of spectral theory of problem T 407
1 An analog of problem T in a finite three dimensional
simply connected region 407
2 Analog of problem T in a three dimensional cylindrical
domain 410
3 Tricomi s problem with a spectral parameter 413
4 Another problem for Eq. (2.176) 414
5 A multi dimensional analog of the problem considered in
the previous section 420
Chapter V. Second order nonlinear equations 423
§ 1. Structural properties of solutions of some classes of nonlinear
partial differential equations 423
1 General remarks 423
CONTENTS xi
2 The case of one equation 423
3 The case of a system of equations 425
§2. Some simple examples 426
1 The Reid and Burt equation 426
2 Example of an equation of elliptic type. Dirichlet s
problem 428
3 Neumann s problem 430
4 The second special case of Eq. (5.21) 432
5 An example of an equation of mixed type 434
§3. Waves in a liquid of variable density 442
1 Derivation of the basic Dubreil Jacotin equation 442
2 Formulation of the general problem 445
3 The case of a rectilinear strip. Reduction to Dirichlet s
problem for the Helmholtz equation 446
4 Construction of the solution 450
§4. Equations of the gravitational field 452
1 A version of the Maxwell Einstein equations for an
axisymmetric stationary gravitational field 452
2 Construction of classes of solutions of Maxwell Einstein
equations 453
3 Ernst s version of the Maxwell Einstein equations and the
construction of its solutions 455
4 Remarks in connection with boundary value problems 460
§5. Singular mappings of two dimensional Riemannian manifolds 461
1 The concept of singular mappings of two dimensional
Riemannian manifolds 461
2 The basic property of singular mappings 463
3 The linear boundary value problem of singular mappings
and its reduction to a nonlinear boundary value problem of
the Laplace equation 464
References 468
Subject index 494
|
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| author | Bicadze, Andrej V. |
| author_facet | Bicadze, Andrej V. |
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| bvnumber | BV002425005 |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:45:02Z |
| institution | BVB |
| isbn | 2881246621 |
| language | English |
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| physical | XI, 504 S. |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Gordon and Breach |
| record_format | marc |
| series | Advanced studies in contemporary mathematics |
| series2 | Advanced studies in contemporary mathematics |
| spelling | Bicadze, Andrej V. Verfasser aut Nekotorye klassy uravnenij v častnych proizvodnych Some classes of partial differential equations A. V. Bitsadze New York [u.a.] Gordon and Breach 1988 XI, 504 S. txt rdacontent n rdamedia nc rdacarrier Advanced studies in contemporary mathematics 4 Aus d. Russ. übers. Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Advanced studies in contemporary mathematics 4 (DE-604)BV000600809 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001573989&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bicadze, Andrej V. Some classes of partial differential equations Advanced studies in contemporary mathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | Some classes of partial differential equations |
| title_alt | Nekotorye klassy uravnenij v častnych proizvodnych |
| title_auth | Some classes of partial differential equations |
| title_exact_search | Some classes of partial differential equations |
| title_full | Some classes of partial differential equations A. V. Bitsadze |
| title_fullStr | Some classes of partial differential equations A. V. Bitsadze |
| title_full_unstemmed | Some classes of partial differential equations A. V. Bitsadze |
| title_short | Some classes of partial differential equations |
| title_sort | some classes of partial differential equations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001573989&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000600809 |
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