METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS

Transcription

1 METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS V.I. Agoshkov, P.B. Dubovski, V.P. Shutyaev CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING

2 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 _ Point sets. Class of functions C P (Q), C P (Q) Point Sets Classes C(Q),(?(Q) Examples from the theory of linear spaces Normalised space The space of continuous functions C(Q) Spaces C*(fi) Space L p (Q) L 2 (Q) Space. Orthonormal systems " Hilbert spaces Space L,(fi) Orthonormal systems Linear operators and functional Linear operators and functionals Inverse operators Adjoint, symmetric and self-adjoint operators Positive operators and energetic space 16 < Linear equations ; Eigenvalue problems.' Generalized derivatives. Sobolev spaces Generalized derivatives Sobolev spaces The Green formula ' Main equations and problems of mathematical physics Main equations of mathematical physics Laplace and Poisson equations Equations of oscillations Helmholtz equation Diffusion and heat conduction equations Maxwell and telegraph equations Transfer equation Gas- and hydrodynamic equations : Classification of linear differential equations 29 vii

3 3.2. Formulation of the main problems of mathematical physics Classification of boundary-value problems The Cauchy problem The boundary-value problem for the elliptical equation Mixed problems Validity of formulation of problems. Cauchy-Kovalevskii theorem Generalized formulations and solutions of mathematical physics problems Generalized formulations and solutions of elliptical problems Generalized formulations and solution of hyperbolic problems The generalized formulation and solutions of parabolic problems Variational formulations of problems Variational formulation of problems in the case of positive definite operators Variational formulation of the problem in the case of positive operators Variational formulation of the basic elliptical problems Integral equations Integral Fredholm equation of the 1st and 2nd kind Volterra integral equations Integral equations with a polar kernel Fredholm theorem Integral equation with the Hermitian kernel 52 Bibliographic commentary METHODS OF POTENTIAL THEORY 56 Main concepts and designations h Introduction Fundamentals of potential theory Additional information from mathematical analysis Main orthogonal coordinates Main differential operations of the vector field Formulae from the field theory Main properties of harmonic functions Potential of volume masses or charges Newton (Coulomb) potential The properties of the Newton potential Potential of a homogeneous sphere Properties of the potential of volume-distributed masses Logarithmic potential Definition of the logarithmic potential The properties of the logarithmic potential The logarithmic potential of a circle with constant density The simple layer potential Definition of the simple layer potential in space The properties of the simple layer potential The potential of the homogeneous sphere The simple layer potential on a plane 66 vni

4 2.5. Double layer potential Dipole potential The double layer potential in space and its properties The logarithmic double layer potential and its properties Using the potential theory in classic problems of mathematical physics Solution of the Laplace and Poisson equations Formulation of the boundary-value problems of the Laplace equation Solution of the Dirichlet problem in space Solution of the Dirichlet problem on a plane Solution of the Neumann problem Solution of the third boundary-value problem for the Laplace equation Solution of the boundary-value problem for the Poisson equation The Green function of the Laplace operator The Poisson equation The Green function Solution of the Dirichlet problem for simple domains Solution of the Laplace equation for complex domains Schwarz method The sweep method Other applications of the potential method Application of the potential methods to the Helmholtz equation Main facts, Boundary-value problems for the Helmholtz equations Green function Equation Av-Xv Non-stationary potentials ; Potentials for the one-dimensional heat equation Heat sources in multidimensional case The boundary-value problem for the wave equation 90 Bibliographic commentary 92, 3. EIGENFUNCTION METHODS 1 94 Main concepts and notations.: Introduction Eigenvalue problems Formulation and theory Eigenvalue problems for differential operators Properties of eigenvalues and eigenfunctions Fourier series Eigenfunctions of some one-dimensional problems Special functions Spherical functions Legendre polynomials Cylindrical functions Chebyshef, Laguerre and Hermite polynomials Mathieu functions and hypergeometrical functions 109

5 4. Eigenfunction method General scheme of the eigenfunction method The eigenfunction method for differential equations of mathematical physics Ill 4.3. Solution of problems with nonhomogeneous boundary conditions Eigenfunction method for problems of the theory of electromagnetic phenomena The problem of abounded telegraph line Electrostatic field inside an infinite prism Problem of the electrostatic field inside a cylinder The field inside a ball at a given potential on its surface The field of a charge induced on a ball Eigenfunction method for heat conductivity problems Heat conductivity in a bounded bar Stationary distribution of temperature in an infinite prism Temperature distribution of a homogeneous cylinder Eigenfunction method for problems in the theory of oscillations Free oscillations of a homogeneous string Oscillations of the string with a moving end Problem of acoustics of free oscillations of gas Oscillations of a membrane with a fixed end Problem of oscillation of a circular membrane 128 Bibliographic commentary METHODS OFJNTEGRAL TRANSFORMS 130 Main concepts and definitions Introduction Main integral transformations'.: Fourier transform The main properties of Fourier transforms Multiple Fourier transform Laplace transform! Laplace integral.' The inversion formula for the Laplace transform Main formulae and limiting theorems Mellin transform Hankel transform Meyer transform Kontorovich-Lebedev transform Meller-Fock transform Hilbert transform Laguerre and Legendre transforms Bochner and convolution transforms, wavelets and chain transforms Using integral transforms in problems of oscillation theory Electrical oscillations Transverse vibrations of a string 143

6 3.3. Transverse vibrations of an infinite circular membrane Using integral transforms in heat conductivity problems Solving heat conductivity problems using the Laplace transform Solution of a heat conductivity problem using Fourier transforms Temperature regime of a spherical ball Using integral transformations in the theory of neutron diffusion The solution of the equation of deceleration of neutrons for a moderator of infinite dimensions The problem of diffusion of thermal neutrons Application of integral transformations to hydrodynamic problems A two-dimensional vortex-free flow of an ideal liquid The flow of the ideal liquid through a slit Discharge of the ideal liquid through a circular orifice Using integral transforms in elasticity theory Axisymmetric stresses in acylinder Bussinesq problem;for the half space Determination of stresses in a wedge Using integral transforms in coagulation kinetics Exact solution of the coagulation equation Violation of the mass conservation law 161 Bibliographic commentary.; METHODS OF DISCRETISATION OF MATHEMATICAL PHYSICS PROBLEMS 163 Main definitions and notations Introduction ' Finite-difference methods The net method Main concepts and definitions of the method General definitions of the net method. The convergence theorem 170, The net method for partial differential equations The method of arbitrary lines : The method of arbitrary lines for parabolic-type equations The method of arbitrary lines for hyperbolic equations The method of arbitrary lines for elliptical equations The net method for integral equations (the quadrature method) Variational methods Main concepts of variational formulations of problems and variational methods Variational formulations of problems Concepts of the direct methods in calculus of variations The Ritz method The classic Ritz method The Ritz method in energy spaces Natural and main boundary-value conditions The method of least squares 195 XI

7 3.4. Kantorovich, Courant and Trefftz methods The Kantorovich method 1% Courant method 1% Trefftz method, Variational methods in the eigenvalue problem Projection methods TheBubnov-Galerkin method v TheBubnov-Galerkin method (a general case) TheBubnov-Galerkin method (i4=>\ 0 +J5) The moments method Projection methods in the Hilbert and Banach spaces The projection method in the Hilbert space The Galerkin-Petrov method The projection method in the Banach space The collocation method Main concepts of the projection-grid methods Methods of integral identities The main concepts of the method The method of Marchuk's integral identity Generalized formulation of the method of integral identities Algorithm of constructing integral identities The difference method of approximating the integral identities The projection method of approximating the integral identities Applications of the methods of integral identities in mathematical physics problems The method of integral identities for the diffusion equation The solution of degenerating equations The method of integral identities for eigenvalue problems 221 Bibliographic Commentary SPLITTING METHODS Introduction Information from the theory of evolution equations and difference schemes Evolution equations The Cauchy problem The nonhomogeneous evolution equation Evolution equations with bounded operators Operator equations in finite-dimensional spaces The evolution system Stationarisation method Concepts and information from the theory of difference schemes Approximation Stability Convergence : The sweep method Splitting methods..: 242 XII

8 3.1. The method of component splitting (the fractional step methods) The splitting method based on implicit schemes of the first order of accuracy The method of component splitting based on the Cranck-Nicholson schemes Methods of two-cyclic multi-component "splitting The method of two-cyclic multi-component splitting Method of two-cyclic component splitting for quasi-linear problems The splitting method with factorisation of operators The implicit splitting scheme with approximate factorisation of the operator The stabilisation method (the explicit-implicit schemes with approximate factorisation of the operator) The predictor-corrector method The predictor-corrector method. The case A =A t +A The predictor-corrector method. Case A= > _ \ The alternating-direction method and the method of the stabilising correction The alternating-direction method The method of stabilising correction Weak approximation method The main system of problems Two-cyclic method of weak approximation The splitting methods - iteration methods of solving stationary problems The general concepts of the theory of iteration methods Iteration algorithms Splitting methods for applied problems of mathematical physics Splitting methods of heat conduction equations The fractional step method Locally one-dimensional schemes Alternating-direction schemes A Splitting methods for hydrodynamics problems Splitting methods for Navier-Stokes equations The fractional steps method for the shallow water equations Splitting methods for the model of dynamics of sea and ocean flows The non-stationary model of dynamics of sea and ocean flows The splitting method '. 270 Bibliographic Commentary METHODS FOR SOLVING NON-LINEAR EQUATIONS 273 Main concepts and Definitions Introduction Elements of nonlinear analysis Continuity and differentiability of nonlinear mappings Main definitions 276 xiii

9 Derivative and gradient of the functional Differentiability according to Frechet Derivatives of high orders and Taylor series Adjoint nonlinear operators Adjoint nonlinear operators and their properties Symmetry and skew symmetry Convex functionals and monotonic operators Variational method of examining nonlinear equations Extreme and critical points of functionals The theorems of existence of critical points Main concept of the variational method The solvability of the equations with monotonic operators Minimising sequences Minimizing sequences and their properties Correct formulation of the minimisation problem The method of the steepest descent Non-linear equation and its variational formulation Main concept of the steepest descent methods Convergence of the method The Ritz method Approximations and Ritz systems Solvability of the Ritz systems Convergence of the Ritz method The Newton-Kantorovich method Description of the Newton iteration process The convergence of the Newton iteration process The modified Newton method The Galerkin-Petrov method for non-linear equations Approximations and Galerkin systems Relation to projection methods Solvability of the Galerkin systems The convergence of the Galerkin-Petrov method Perturbation method Formulation of the perturbation algorithm Justification of the perturbation algorithms Relation to the method of successive approximations Applications to some problem of mathematical physics The perturbation method for a quasi-linear problem of non-stationary heat conduction The Galerkin method for problems of dynamics of atmospheric processes The Newton method in problems of variational data assimilation 308 Bibliographic Commentary 311 Index 317 xiv