METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS i www.pdfgrip.com ii www.pdfgrip.com METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS V.I Agoshkov, P.B Dubovski, V.P Shutyaev CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING iii www.pdfgrip.com Published by Cambridge International Science Publishing 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 www.pdfgrip.com 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, and were written by V.I Agoshkov, chapters and by P.B Dubovski, and chapters and 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 v www.pdfgrip.com vi www.pdfgrip.com Contents PREFACE MAIN PROBLEMS OF MATHEMATICAL PHYSICS Main concepts and notations Introduction Concepts and assumptions from the theory of functions and functional analysis 2.1 Point sets Class of functions C p (Ω), C p (Ω) 2.1.1 Point Sets 2.1.2 Classes Cp(Ω), Cp( Ω ) 2.2 Examples from the theory of linear spaces 2.2.1 Normalised space 3 5 2.2.2 The space of continuous functions C( Ω ) 2.2.3 Spaces Cλ (Ω) 2.2.4 Space Lp(Ω) 2.3 L2(Ω) Space Orthonormal systems 2.3.1 Hilbert spaces 2.3.2 Space L2(Ω) 11 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 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 vii www.pdfgrip.com 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 METHODS OF POTENTIAL THEORY 56 Main concepts and designations 56 Introduction 57 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 viii www.pdfgrip.com 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 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 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 = 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 EIGENFUNCTION METHODS 94 Main concepts and notations 94 Introduction 94 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 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 ix www.pdfgrip.com Methods for Solving Mathematical Physics Problems N R − ∂R ∂R1 N − div L gradR1 = − C − 1 , ∂t β β β ∂t ∂R1 R1 t =0 = 0, R1 = 0, γ = 0, γ1 ∂n N R − ∂R N ∂R ∂R2 N − div L gradR2 = − C − 1 − C ' R1 , ∂t ∂ ∂t β β β t β ∂R2 R2 t = = 0, R2 = 0, γ =0 γ1 ∂n ( ( ) ) and so on Computing N corrections R i , i = 1, ,N, we find the approximation of the N-th order to R using the equation: N R( N ) = R0 + ∑ ε i Ri , i =0 where according to the theorem 28 R − R( N ) Y ≤ c ε n +1 , c=const>0 8.2 The Galerkin method for problems of dynamics of atmospheric processes Let S be a sphere with radius r We examine the problem of the twodimensional equation of a barotropic atmosphere on a sphere in the form [23] ∂φ s + v ( −∆ ) φ + J ( ∆ −1φ, φ ) = f ∈ ( 0, T ) , φ ( ) = u , (54) ∂t where J ( v, w ) = ∂v ∂w ∂v ∂w − , r ∂λ ∂µ ∂µ ∂λ φ = φ (t , λ,µ), µ=sinψ, (λ,ψ) ∈ S , ≤ λ ≤ 2π, − π / ≤ ψ ≤ π / 2, ∆ is the Laplace-Beltrami operator on the sphere ∂2 ∂ ∂ + (1 − µ ) (55) 2 r − µ ∂λ ∂µ ∂µ Here ϕ = ϕ(t,λ,µ ) is the function of the vortex, λ is the longitude, ψ is the latitude, t∈[0,T], v = const>0, s≥1, the member v(−∆) sϕ describes the turbulent viscosity, f(λ,t,µ ) is the external source of vorticity, u=u(λ,µ ) is the function of the initial condition In this model, the atmosphere is treated as a layer of an incompressible fluid of constant density whose thickness is small in comparison with the horizontal scale of motion Regardless of relative simplicity, this equation also takes into account the important dynamic processes as non-linear interaction and wave dispersion The fine-scale motion of the atmosphere within this model is taken into account and by means of the turbulent member and the external source of vorticity ∆= 306 www.pdfgrip.com Methods for Solving Non-Linear Equations When s = 1, equation (54) is derived from the conventional NavierStokes equations on the sphere ‘Artificial viscosity’, when s > is often used to prove the theorems of existence and uniqueness, and also in the numerical solution of the problem We introduce H = L ( S ) − the Hilbert space of real-valued functions, defined on S, square-integrable and orthogonal to the constant, with the normal scalar product (·,·) and the norm ||·||=(·,·) 1/2 The Laplace“Beltrami operator will be regarded as the operator acting from H into H with the domain of definition D(A)={u∈H:∆u∈H} Using the powers of the Laplace operator, we can introduce the Sobolev Ο spaces Η γ (S), γ∈R, with the scalar product and the norm: ( u, v ) H ( S ) = ( (−∆ ) γ/2 γ u, ( −∆ ) γ/2 ∞ v) = ∑ Λ γj u j v j , j =1 1/ (56) ∞ u = ∑ Λ γj u 2j , H γ (S) (S) j =1 where Λ j are the eigenvalues of the operator −∆, corresponding to the u =( u , u )Η = ( −∆ ) 1/ Ο γ/2 γ eigenfunctions ω j , u j = (u,ω j ) Here D ( ∆ ) = H ( S ), H = H ( S ) We define the operator A using the equation Aφ=v ( −∆ ) φ; s (57) o it acts in H with the domain of definition D(A)= H 2s (S) For γ∈R we introduce dφ the spaces X γ = H 2γs , Y γ = L2 ( 0, T ; X γ ) , and W γ = φ ∈ Y γ +1/2 : ∈Y γ −1/ with dt the norms 1/ dφ T + φ φ Y γ = ∫ φ X γ dt , φ W γ = dt Y γ−1/ 0 For ϕ∈W γ we defined a non-linear operator F(ϕ) 1/ Y γ+1/ F ( φ ) = J ( ∆ −1φ, φ ) It is well known that if s > 1, γ > 1/(2s) or 0≤γ≤(s–1)/(2s) or s>1, then the operator F acts from Y γ + 1/2 into Y γ –1/2 with the domain of definition D(F) = W γ , it is continuously differentiable according to Frechét and F ' ( φ ) Y γ+1/ →Y γ−1/ ≤ c3 φ W γ , c3 = const > Taking this into account, the problem (54) can be written in the operator form dφ + Aφ + F ( φ ) = f , t ∈ ( 0, T ) , φ ( ) = u (58) dt It is an abstract non-linear evolution equation This problem can be also written in the form A ϕ=f, selecting as A the operator Aϕ = 307 www.pdfgrip.com dϕ + Aϕ + F (ϕ) dt Methods for Solving Mathematical Physics Problems acting from Y γ +1/2 into D(A )={ϕ∈W γ :ϕ(0)=u} Y γ –1/2 with the domain of definition −s It is well-known that at f∈Y –1/2 =L (0,T; H ), s ≥ 1, u∈ L (S) there is a unique solution ϕ∈W of the problem (58) We examine the Galerkin method to solve the problem (54) As basis functions we examine the finite set of the eigenfunctions {ω j } j =1, N of the Laplace–Beltrami operator The approximate solution of equation (58) will be determined in the form N φN (t ) = ∑ φN j (t ) ω j j =1 (59) Introducing the projection operator P N using the equation N PN ξ = ∑ ξ j ω j , j =1 ξ j = ( ξ, ω j ) and applying it to equation (58), we get ∂φ N + PN J ( ∆ −1 φ N , φ N ) = v∆φ N + PN f (60) ∂t This gives the system of ordinary differential equations for determining the functions ϕ Nj (t) N ( ) φ′Nj + vΛ sj φ Nj + ∑ J ( ∆ −1ωi ,ωk ) ,ω j φ Nj φ Nk = ( f ,ω j ) , i ,k =1 (61) j = 1, , N φ Nj ( ) = ( u ,ω j ) The existence of the solution of the system (61) follows from the theory of ordinary differential equations and a priori estimates On the basis of solvability of the system (61) we establish the existence and uniqueness of the solution of the initial problem (54) 8.3 The Newton method in problems of variational data assimilation At present, because of the investigations of global changes, it is important to examine the problem of obtaining and rational application of the results of measurements for retrospective analysis in various areas of knowledge The mathematical model of the problem may be formulated as a problem of the collection and processing of multi-dimensional (including the dependence of the time and spatial variables) data representing one of the optimal control problems Let us assume that we examine some physical process whose mathematical model is written in the form of a non-linear evolution problem ∂φ = F ( φ ) , t ∈ ( 0, T ) , φ t = = u , (62) ∂t where ϕ=ϕ(t) is the unknown function belonging for each t to the Hilbert space X, u∈X, F is the non-linear operator acting from X into X Let Y = L (0,T; X), (·,·) L2 (0,T ; X ) =(·,·), ||·||=(·,·) 1/2 We introduce the functional 308 www.pdfgrip.com Methods for Solving Non-Linear Equations α 2 u − uobs X + ∫ Cφ − φobs X dt , (63) 20 where α = const ≥ 0, u obs ∈X, ϕ οbs ∈Y are the given functions (the results of observation), Y obs is a sub-space of Y, C:Y→Y obs is the linear operator We examine the following problem of data assimilation in order to restore the initial condition: find u and ϕ such that ∂φ = F ( φ ) , t ∈ ( 0, T ) , φ|t =0 = u , S ( u ) = inf S ( v ) (64) v ∂t The necessary optimality condition reduces the problem (64) to the system ∂φ = F ( φ ) , t ∈ ( 0, T ) , φ t = = u , (65) ∂t ∗ ∗ ∂φ − − ( F ' ( φ ) ) φ∗ = −C ∗ ( Cφ − φ obs ) , t ∈ ( 0, T ) , φ∗ t =T = 0, (66) ∂t α ( u − uobs ) − φ∗ t = = (67) with the unknown ϕ,ϕ*, u, where (F'(ϕ))* is the operator adjoint to the Frechét derivative of the operator F, and C* is the operator adjoint to C Assuming that there is a solution of the problem (65)–(67), we examine the Newton method to find this solution The system (65)–(67) with three unknown ϕ,ϕ*,u can be regarded as the operator equation of the type F (U ) = 0, (68) where U = (ϕ,ϕ*,u) To use the Newton method, we must calculate F'(U) It is assumed that the original operator F is differentiable twice continuously according to Frechét Consequently, the Newton method T S (u ) = U n +1 = U n − F ' (U n ) F (U n ) , U n = ( φ n ,φ∗n , un ) (69) consists of the following steps We find Vn=[F'(U n)] –1F'(U n) as a solution of the problem F'(Un) V n=F(U n) at V n = (ψ n ,ψ*n ,v n ) ∂ψ n ∂φ − F ' ( φ n ) ψ n = n − F ( φ n ) , ψ n t = = + φ n t = −un , (70) ∂t ∂t ∂ψ∗ ∗ − n − ( F ' ( φ n ) ) ψ∗n = p1n , ψ∗n t =T = φ∗n t =T , (71) ∂t −1 αvn − ψ ∗n t =0 = α ( un − uobs ) − φ∗n n ∗ ∗ where p1 = ( F " ( φ n ) ψn ) φ n − C Cψn − ∗ Assume U n + t =0 , (72) ∂φ∗n ∗ − ( F ' ( φ n ) ) φ∗n + C ∗ ( Cφ n − φ obs ) ∂t =U n −V n , i.e φ n +1 = φ n − ψ n , φ∗n +1 = φ ∗n − ψ∗n , un +1 = un − (73) Since U n+1 =U n–V n , then the two steps (70)–(73) can be reformulated as follows: at given φ n , φ∗n , un find φ n +1 , φ ∗n +1 , un +1 such that 309 www.pdfgrip.com Methods for Solving Mathematical Physics Problems ∂φ n +1 − F ' ( φ n ) φ n +1 = F ( φ n ) − F ' ( φ n ) φ n , φ n +1 t = = un +1 , ∂t ∂φ∗ ∗ − n +1 − ( F ' ( φ n ) ) φ∗n +1 = p2n , φ∗n +1 t =T = 0, ∂t α ( un +1 − uobs ) − φ∗n +1 t = = 0, (74) (75) (76) where p2n = ( F " ( φ n )( φ n +1 − φ n ) ) φ∗n − C ∗ ( Cφ n +1 − φ obs ) We fix the point ϕ ∈Y, the real number R > and examine the ball S R(ϕ ) ={ϕ∈Y:||ϕ=ϕ || ≤ R} It is assumed that the initial mathematical model satisfies at all ϕ∈S R (ϕ ) the following conditions: 1) the solution of the problem ∂ψ − F ' ( φ ) ψ = f , ψ t =0 = v ∂t satisfies the inequality ψ ≤ c1 ( f + v X ) , c1 = c1 ( R,φ ) > 0; (77) 2) the solution of the adjoint problem ∗ ∂ψ∗ − − ( F ' ( φ ) ) ψ∗ = p, ψ∗ t =T = g ∂t satisfies ∗ ψ∗ + ψ∗ ≤ c1∗ ( p + g ), c1∗ = c1∗ ( R , φ ) > 0; (78) 3) operator F is three time continuously differentiable according to Frechét, and F " ( φ ) ≤ c2 , F "' ( φ ) ≤ c3 , ck = ck ( R, φ ) > 0, k = 1, (79) t =0 X X Comment For a bi-linear operator F constant c does not depend on R, ϕ and c ≡0 We find the solution of the problem (65)–(67) in the ball Sr = {( φ, φ , u ) : φ − φ ∗ } + φ ∗ + u − u0 x ≤ r , u0 ∈ X , r = ( c2−1 , R ) In the conditions of complete observation (C≡E), it holds Theorem 29 Let u ∈X, ϕ ∈Y, R > 0, φ*0 = B ( c2 + c3r ) r B η1 + ≤ r, (80) where η= ∂φ − F ( φ0 ) + φ ∂t t =0 −u X + φ0 − φ obs + α u0 − uobs X , B = max ( β1 ,β ,β ) , β1 = α −1 ( 2c1c1∗ + 2c12 c1∗ + 4c12 c1∗2 ) + c1 + 2c1c1∗ , β = α −1 ( c1∗ + c1c1∗ + 2c1c1∗2 ) + c1∗ , β = α −1 (1 + c1 + 2c1c1∗ ) 310 www.pdfgrip.com Methods for Solving Non-Linear Equations Consequently, (65)–(67) has a unique solution ϕ, ϕ*, u in the ball S r Starting from ϕ , ϕ*0, u 0, the Newton method converges to ϕ, ϕ*, u The following estimate of the convergence rate holds: ( h/2 ) ≤ Bη − ( h / 2) 2n −1 ∗ ∗ n φ − φ n + φ − φ + u − un X where h = B η(c +c r)