Methods of Mathematical Physics: Classical and Modern

Description

This textbook provides a thorough overview of mathematical physics, highlighting classical topics as well as recent developments. Readers will be introduced to a variety of methods that reflect current trends in research, including the Bergman kernel approach for solving boundary value and spectral problems for PDEs with variable coefficients. With its careful treatment of the fundamentals as well as coverage of topics not often encountered in textbooks, this will be an ideal text for both introductory and more specialized courses. The first five chapters present standard material, including the classification of PDEs, an introduction to boundary value and initial value problems, and an introduction to the Fourier method of separation of variables. More advanced material and specialized treatments follow, including practical methods for solving direct and inverse Sturm-Liouville problems; the theory of parabolic equations, harmonic functions, potential theory, integral equations and the method of non-orthogonal series. Methods of Mathematical Physics is ideal for undergraduate students and can serve as a textbook for a regular course in equations of mathematical physics as well as for more advanced courses on selected topics. Preface Contents 1 Introduction 2 Classification of PDEs 2.1 Classification of Linear PDEs 2.2 Canonical Forms of the Equations 2.2.1 Reduction of Hyperbolic Equations 2.2.2 Reduction of Parabolic Equations 2.2.3 Reduction of Elliptic Equations 2.3 Classification in Case of n Variables 3 Models of Mathematical Physics 3.1 Equation of Small Vibrations of a String 3.2 Energy of Free Vibrations of the String 3.3 Longitudinal Vibrations of a Rod 3.4 Equation of Vibrations of a Membrane 3.5 Equations of Hydrodynamics 3.6 Sound Propagation and Equations of Acoustics 3.7 Heat Equation 3.8 Diffusion Equation 3.9 Problems Reducing to Elliptic Equations 3.10 Helmholtz Equation 3.11 Equation of Electric and Electromagnetic Oscillations 3.12 Schrödinger Equation 4 Boundary Value Problem Statements for Partial Differential Equations 4.1 Cauchy's Problem Statement and Boundary Conditions. Equations of Normal Type 4.2 Boundary Value Problems for the Wave Equation 4.3 Boundary Value Problems for the Heat Equation 4.4 Boundary Value Problems for the Poisson and Laplace Equations 4.5 Boundary and Initial Conditions for the Telegraph Equation 4.6 Well Posedness of Problems of Mathematical Physics 4.7 Notion of Generalized Solutions 5 Cauchy Problem for Hyperbolic Equations 5.1 Cauchy Problem for the One-Dimensional Wave Equation 5.2 Physical Meaning of d'Alembert Formula 5.3 Uniqueness Theorem for the Wave Equation 5.4 Solution of Cauchy's Problem: Kirchhoff's Formula 5.5 Method of Descent: Poisson and d'Alembert Formulas 5.6 A Closer Look at the Formulas for Solutions of Cauchy's Problems: Wave Propagation—Stability of Solution of Cauchy's Problem—Wave Diffusion 6 Fourier Method for the Wave Equation 6.1 Uniqueness Theorem for the First Boundary Value Problem for the Wave Equation 6.2 Fourier Method for a Fixed String 6.3 General Scheme of the Fourier Method 6.4 Properties of Eigenfunctions and Eigenvalues 6.5 Free Vibrations of a Rectangular Membrane 6.6 Solution of a Non-homogeneous Problem 6.7 Free Vibrations of a Circular Membrane 6.8 Bessel Equation and Bessel Functions 6.9 Properties of Bessel Functions 6.10 Expansion in Series of Bessel Functions 6.11 Free Vibrations of a Circular Membrane (Continuation) 7 Sturm-Liouville Problems 7.1 Polya's Factorization, Abel's Formula 7.2 Spectral Parameter Power Series 7.3 Some Properties of Formal Powers 7.4 SPPS Method for Solving Sturm-Liouville Problems 7.5 Transmutation Operators 7.6 NSBF for Solutions 7.7 Jost Solutions 7.8 NSBF method 7.9 Numerical Example. NSBF vs. SPPS 7.10 Scattering Problem on the Line 7.11 Notion of Inverse Spectral Problems 7.12 Direct Method of Solution of the Inverse Sturm-Liouville Problem 7.12.1 Numerical Realization 7.13 Direct Method of Solution of the Inverse Scattering Problem 7.13.1 Numerical Realization 7.14 Inverse Scattering Transform Method 8 Boundary Value Problems for the Heat Equation 8.1 Maximum and Minimum Principle for the Heat Equation 8.2 Uniqueness Theorem for Unbounded Domains (on the Example of a Line) 8.3 Heat Transfer in an Infinite Rod 8.4 Corollaries from the Poisson Formula 8.5 Heat Flow in a Semi-Infinite Rod 8.6 Heat Flow on a Finite Interval 8.7 Heat Source Function 9 Harmonic Functions and Their Properties 9.1 Definition of Harmonic Function, Fundamental Solution of the Laplace Equation 9.2 Green's Function and Integral Representation of Functions from the Class C2(V) 9.3 Properties of Harmonic Functions 9.4 Behavior of Harmonic Functions at Infinity 10 Boundary Value Problems for the Laplace Equation 10.1 Statement of Interior and Exterior Boundary Value Problems 10.2 Uniqueness Theorems for Dirichlet and Neumann Problems 10.3 Green's Function of the Dirichlet Problem 10.4 Properties of Green's Function 10.5 Physical Meaning of Green's Function 10.6 Method of Electrostatic Images and Solution of the Dirichlet Problem for a Ball 10.7 Solution of the Dirichlet Problem in a Disk 10.8 Solution of the Dirichlet Problem for a Half-Plane 10.9 Solution of the Dirichlet Problem for a Half-Space 11 Potential Theory 11.1 Theorems on the Volume Potential 11.2 Lyapunov Surfaces and Their Properties 11.3 Solid Angle of Lyapunov Surface 11.4 Surface Potentials on Lyapunov Surfaces 11.5 Calculation of the Gauss Integral 11.6 Jump Discontinuity of the Double Layer Potential 11.7 Normal Derivative of the Single Layer Potential 11.8 Reduction of Boundary Value Problems for the Laplace Equation to Integral Equations 12 Elements of Theory of Integral Equations 12.1 Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L 2 left parenthesis a comma b right parenthesis) /StPNE pdfmark [/StBMC pdfmarkL2( a,b) ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and Its Properties 12.2 Fredholm Operator and Its Iterated Kernels 12.3 Method of Successive Approximations 12.4 Notion of the Resolvent of the Integral Equation 12.5 Volterra Integral Equation 12.6 Integral Equations with Degenerate Kernels 12.7 Integral Equations in a General Case 12.8 Regular and Characteristic Values 12.9 Adjoint Integral Equation 12.10 Properties of the Adjoint Operator 12.11 Fredholm Theorems 12.12 Several Independent Variables 12.13 Equations with a Weak Singularity 12.14 Continuity of Solutions of Integral Equations 12.14.1 Continuous in the Whole Kernels 12.14.2 Weakly Singular Equations 12.15 Symmetric Integral Equations 12.16 Hilbert–Schmidt Theorem 12.17 Solution of Symmetric Integral Equations 12.18 Bilinear Series 12.19 Bilinear Series for Iterated Kernels 12.20 Resolvent of a Symmetric Kernel 12.21 Extremal Properties of Eigenvalues and Eigenfunctions 13 Solution of Boundary Value Problems for the Laplace Equation 13.1 Problems D+ and N- 13.2 Problem N+ 13.3 Problem D- 14 Helmholtz Equation 14.1 Definition and Relations with Time-Dependent Models 14.2 Fundamental Solutions 14.3 Integral Representation for Solutions 14.4 Interior Boundary Value Problems 14.5 Maximum and Minimum Principles 14.6 Exterior Boundary Value Problems: Uniqueness 14.6.1 Existence 15 Method of Non-orthogonal Series 15.1 Complete Systems of Solutions 15.1.1 Spherical Wave Functions 15.1.2 Fundamental Solutions 15.1.3 Plane Waves 15.1.4 Linear Independence 16 Bergman Kernel Approach 16.1 Fundamental Solutions 16.2 Green's and Neumann's Functions 16.3 Bergman's Kernel 16.4 Energy Integral and Scalar Product 16.5 Complete Systems of Solutions and Construction of Bergman's Kernel 16.6 Reproducing Kernels for Arbitrary q 16.7 Construction of Complete Systems of Solutions Bibliography Index