Methods for Solving Mathematical Physics Problems

Front Cover
Cambridge Int Science Publishing, 2006 - Science - 320 pages
1 Review
The aim of the book is to present to a wide range of readers (students, postgraduates, scientists, engineers, etc.) basic information on one of the directions of mathematics, methods for solving mathematical physics problems. The authors have tried to select for the book methods that have become classical and generally accepted. However, some of the current versions of these methods may be missing from the book because they require special knowledge. The book is of the handbook-teaching type. On the one hand, the book describes the main definitions, the concepts of the examined methods and approaches used in them, and also the results and claims obtained in every specific case. On the other hand, proofs of the majority of these results are not presented and they are given only in the simplest (methodological) cases. Another special feature of the book is the inclusion of many examples of application of the methods for solving specific mathematical physics problems of applied nature used in various areas of science and social activity, such as power engineering, environmental protection, hydrodynamics, elasticity theory, etc. This should provide additional information on possible applications of these methods. To provide complete information, the book includes a chapter dealing with the main problems of mathematical physics, together with the results obtained in functional analysis and boundary-value theory for equations with partial derivatives.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

33 Transverse vibrations of an infinite circular membrane
146
4 USING INTEGRAL TRANSFORMS IN HEAT CONDUCTIVITY PROBLEMS
147
42 Solution of a heat conductivity problem using Fourier transforms
148
43 Temperature regime of a spherical ball
149
51 The solution of the equation of deceleration of neutrons for a moderator of infinite dimensions
150
6 APPLICATION OF INTEGRAL TRANSFORMATIONS TO HYDRODYNAMIC PROBLEMS
151
62 The flow of the ideal liquid through a slit
152
63 Discharge of the ideal liquid through a circular orifice
153

232 Space L2
11
24 Linear operators and functionals
13
242 Inverse operators
15
244 Positive operators and energetic space
16
245 Linear equations
17
25 Generalized derivatives Sobolev spaces
19
252 Sobolev spaces
20
253 The Green formula
21
3 MAIN EQUATIONS AND PROBLEMS OF MATHEMATICAL PHYSICS
22
311 Laplace and Poisson equations
23
312 Equations of oscillations
24
313 Helmholtz equation
26
315 Maxwell and telegraph equations
27
316 Transfer equation
28
317 Gas and hydrodynamic equations
29
32 Formulation of the main problems of mathematical physics
32
322 The Cauchy problem
33
323 The boundaryvalue problem for the elliptical equation
34
324 Mixed problems
35
33 Generalized formulations and solutions of mathematical physics problems
37
331 Generalized formulations and solutions of elliptical problems
38
332 Generalized formulations and solution of hyperbolic problems
41
333 The generalized formulation and solutions of parabolic problems
43
34 Variational formulations of problems
45
342 Variational formulation of the problem in the case of positive operators
46
343 Variational formulation of the basic elliptical problems
47
35 Integral equations
49
352 Volterra integral equations
50
353 Integral equations with a polar kernel
51
355 Integral equation with the Hermitian kernel
52
BIBLIOGRAPHIC COMMENTARY
54
METHODS OF POTENTIAL THEORY
56
1 INTRODUCTION
57
2 FUNDAMENTALS OF POTENTIAL THEORY
58
213 Formulae from the field theory
59
214 Main properties of harmonic functions
60
22 Potential of volume masses or charges
61
223 Potential of a homogeneous sphere
62
23 Logarithmic potential
63
233 The logarithmic potential of a circle with constant density
64
242 The properties of the simple layer potential
65
243 The potential of the homogeneous sphere
66
25 Double layer potential
67
253 The logarithmic double layer potential and its properties
69
3 USING THE POTENTIAL THEORY IN CLASSIC PROBLEMS OF MATHEMATICAL PHYSICS
70
312 Solution of the Dirichlet problem in space
71
313 Solution of the Dirichlet problem on a plane
72
314 Solution of the Neumann problem
73
315 Solution of the third boundaryvalue problem for the Laplace equation
74
316 Solution of the boundaryvalue problem for the Poisson equation
75
32 The Green function of the Laplace operator
76
323 Solution of the Dirichlet problem for simple domains
77
33 Solution of the Laplace equation for complex domains
78
332 The sweep method
80
4 OTHER APPLICATIONS OF THE POTENTIAL METHOD
81
412 Boundaryvalue problems for the Helmholtz equations
82
413 Green function
84
414 Equation vλv 0
85
42 Nonstationary potentials
86
422 Heat sources in multidimensional case
88
423 The boundaryvalue problem for the wave equation
90
BIBLIOGRAPHIC COMMENTARY
92
EIGENFUNCTION METHODS
94
2 EIGENVALUE PROBLEMS
95
22 Eigenvalue problems for differential operators
98
23 Properties of eigenvalues and eigenfunctions
99
24 Fourier series
100
25 Eigenfunctions of some onedimensional problems
102
3 SPECIAL FUNCTIONS
103
32 Legendre polynomials
105
33 Cylindrical functions
106
34 Chebyshef Laguerre and Hermite polynomials
107
35 Mathieu functions and hypergeometrical functions
109
4 EIGENFUNCTION METHOD
110
42 The eigenfunction method for differential equations of mathematical physics
111
43 Solution of problems with nonhomogeneous boundary conditions
114
5 EIGENFUNCTION METHOD FOR PROBLEMS OF THE THEORY OF ELECTROMAGNETIC PHENOMENA
115
52 Electrostatic field inside an infinite prism
117
54 The field inside a ball at a given potential on its surface
118
55 The field of a charge induced on a ball
120
6 EIGENFUNCTION METHOD FOR HEAT CONDUCTIVITY PROBLEMS
121
62 Stationary distribution of temperature in an infinite prism
122
63 Temperature distribution of a homogeneous cylinder
123
7 EIGENFUNCTION METHOD FOR PROBLEMS IN THE THEORY OF OSCILLATIONS
124
72 Oscillations of the string with a moving end
125
73 Problem of acoustics of free oscillations of gas
126
74 Oscillations of a membrane with a fixed end
127
75 Problem of oscillation of a circular membrane
128
BIBLIOGRAPHIC COMMENTARY
129
4 METHODS OF INTEGRAL TRANSFORMS
130
1 INTRODUCTION
131
2 MAIN INTEGRAL TRANSFORMATIONS
132
211 The main properties of Fourier transforms
133
212 Multiple Fourier transform
134
222 The inversion formula for the Laplace transform
135
24 Hankel transform
136
25 Meyer transform
138
27 MellerFock transform
139
28 Hilbert transform
140
210 Bochner and convolution transforms wavelets and chain transforms
141
3 USING INTEGRAL TRANSFORMS IN PROBLEMS OF OSCILLATION THEORY
143
7 USING INTEGRAL TRANSFORMS IN ELASTICITY THEORY
155
72 Bussinesq problem for the half space
157
73 Determination of stresses in a wedge
158
8 USING INTEGRAL TRANSFORMS IN COAGULATION KINETICS
159
82 Violation of the mass conservation law
161
BIBLIOGRAPHIC COMMENTARY
162
5 METHODS OF DISCRETISATION OF MATHEMATICAL PHYSICS PROBLEMS
163
1 INTRODUCTION
164
2 FINITEDIFFERENCE METHODS
166
212 General definitions of the net method The convergence theorem
170
213 The net method for partial differential equations
173
22 The method of arbitrary lines
182
222 The method of arbitrary lines for hyperbolic equations
184
223 The method of arbitrary lines for elliptical equations
185
23 The net method for integral equations the quadrature method
187
3 VARIATIONAL METHODS
188
312 Concepts of the direct methods in calculus of variations
189
32 The Ritz method
190
322 The Ritz method in energy spaces
192
323 Natural and main boundaryvalue conditions
194
33 The method of least squares
195
34 Kantorovich Courant and Trefftz methods
196
343 Trefftz method
197
35 Variational methods in the eigenvalue problem
199
4 PROJECTION METHODS
201
412 The BubnovGalerkin method A A0 +B
202
42 The moments method
204
43 Projection methods in the Hilbert and Banach spaces
205
432 The GalerkinPetrov method
206
434 The collocation method
208
5 METHODS OF INTEGRAL IDENTITIES
210
52 The method of Marchuks integral identity
211
53 Generalized formulation of the method of integral identities
213
532 The difference method of approximating the integral identities
214
533 The projection method of approximating the integral identities
215
54 Applications of the methods of integral identities in mathematical physics problems
218
542 The solution of degenerating equations
219
BIBLIOGRAPHIC COMMENTARY
223
6 SPLITTING METHODS
224
2 INFORMATION FROM THE THEORY OF EVOLUTION EQUATIONS AND DIFFERENCE SCHEMES
225
212 The nonhomogeneous evolution equation
228
213 Evolution equations with bounded operators
229
22 Operator equations in finitedimensional spaces
231
222 Stationarisation method
232
23 Concepts and information from the theory of difference schemes
233
232 Stability
239
233 Convergence
240
234 The sweep method
241
3 SPLITTING METHODS
242
31 The method of component splitting the fractional step methods
243
32 Methods of twocyclic multicomponent splitting
245
322 Method of twocyclic component splitting for quasilinear problems
246
331 The implicit splitting scheme with approximate factorisation of the operator
247
332 The stabilisation method the explicitimplicit schemes with approximate factorisation of the operator
248
34 The predictorcorrector method
250
342 The predictorcorrector method
251
35 The alternatingdirection method and the method of the stabilising correction
252
352 The method of stabilising correction
253
36 Weak approximation method
254
37 The splitting methods iteration methods of solving stationary problems
255
372 Iteration algorithms
256
4 SPLITTING METHODS FOR APPLIED PROBLEMS OF MATHEMATICAL PHYSICS
257
41 Splitting methods of heat conduction equations
258
421 Locally onedimensional schemes
259
42 Splitting methods for hydrodynamics problems
262
422 The fractional steps method for the shallow water equations
263
43 Splitting methods for the model of dynamics of sea and ocean flows
268
432 The splitting method
270
BIBLIOGRAPHIC COMMENTARY
272
7 METHODS FOR SOLVING NONLINEAR EQUATIONS
273
1 INTRODUCTION
274
2 ELEMENTS OF NONLINEAR ANALYSIS
276
212 Derivative and gradient of the functional
277
213 Differentiability according to Fréchet
278
22 Adjoint nonlinear operators
279
222 Symmetry and skew symmetry
280
24 Variational method of examining nonlinear equations
282
243 Main concept of the variational method
283
25 Minimising sequences
284
252 Correct formulation of the minimisation problem
285
32 Main concept of the steepest descent methods
286
33 Convergence of the method
287
4 THE RITZ METHOD
288
41 Approximations and Ritz systems
289
42 Solvability of the Ritz systems
290
43 Convergence of the Ritz method
291
52 The convergence of the Newton iteration process
292
6 THE GALERKINPETROV METHOD FOR NONLINEAR EQUATIONS
293
62 Relation to projection methods
294
63 Solvability of the Galerkin systems
295
7 PERTURBATION METHOD
296
72 Justification of the perturbation algorithms
299
73 Relation to the method of successive approximations
301
8 APPLICATIONS TO SOME PROBLEM OF MATHEMATICAL PHYSICS
302
82 The Galerkin method for problems of dynamics of atmospheric processes
306
83 The Newton method in problems of variational data assimilation
308
BIBLIOGRAPHIC COMMENTARY
311
References
313
Index
317
Copyright

Common terms and phrases

Bibliographic information