Methods for Solving Mathematical Physics Problems

335 Pages·2006·3.516 MB·English

by  V. I. Agoshkov, P. B. Dubovski, V. P. Shutyaev

METHODS FOR
SOLVING
MATHEMATICAL PHYSICS
PROBLEMS
i
ii
METHODS FOR
SOLVING
MATHEMATICAL PHYSICS
PROBLEMS
V.I. Agoshkov, P.B. Dubovski, V.P. Shutyaev
CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING
iii
Published by
Cambridge International Science Publishing
7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK
http://www.cisp-publishing.com
First published October 2006
© V.I. Agoshkov, P.B. Dubovskii, V.P. Shutyaev
© Cambridge International Science Publishing
Conditions of sale
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording, or any information storage and retrieval
system, without permission in writing from the publisher
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British
Library
ISBN 10: 1-904602-05-3
ISBN 13: 978-1-904602-05-7
Cover design Terry Callanan
Printed and bound in the UK by Lightning Source (UK) Ltd
iv
Preface
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.
Chapters 1, 5 and 6 were written by V.I. Agoshkov, chapters 2 and
4 by P.B. Dubovski, and chapters 3 and 7 by V.P. Shutyaev. Each chapter
contains a bibliographic commentary for the literature used in writing the
chapter and recommended for more detailed study of the individual
sections.
The authors are deeply grateful to the editor of the book G.I.
Marchuk, who has supervised for many years studies at the Institute of
Numerical Mathematics of the Russian Academy of Sciences in the area of
computational mathematics and mathematical modelling methods, for his
attention to this work, comments and wishes.
The authors are also grateful to many colleagues at the Institute for
discussion and support.
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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 Cp(Ω),Cp(Ω) ............................................... 3
2.1.1. Point Sets................................................................................................... 3
2.1.2. Classes Cp(Ω), Cp(Ω) ............................................................................... 4
2.2. Examples from the theory of linear spaces....................................................... 5
2.2.1. Normalised space....................................................................................... 5
Ω
2.2.2. The space of continuous functions C( ) ................................................ 6
2.2.3. Spaces Cλ (Ω)............................................................................................. 6
2.2.4. Space L(Ω)................................................................................................ 7
p
2.3. L(Ω) Space. Orthonormal systems................................................................. 9
2
2.3.1. Hilbert spaces ............................................................................................ 9
2.3.2. Space L(Ω)...............................................................................................11
2
2.3.3. Orthonormal systems................................................................................11
2.4. Linear operators and functionals....................................................................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
3.1.2. Equations of oscillations ..........................................................................24
3.1.3. Helmholtz equation...................................................................................26
3.1.4. Diffusion and heat conduction equations ................................................26
3.1.5. Maxwell and telegraph equations.............................................................27
3.1.6. Transfer equation......................................................................................28
3.1.7. Gas- and hydrodynamic equations ..........................................................29
3.1.8. Classification of linear differential equations............................................29
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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...............................................................................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.....................................................60
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
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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 ∆v–λv = 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....................................................... 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
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