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De Gruyter textbook.
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Оглавление
- Preface
- Contents
- 1 Motivation, Derivation of Basic Mathematical Models
- 1.1 Conservation Laws
- 1.1.1 Evolution Conservation Law
- 1.1.2 Stationary Conservation Law
- 1.1.3 Conservation Law in One Dimension
- 1.2 Constitutive Laws
- 1.3 Basic Models
- 1.3.1 Convection and Transport Equation
- 1.3.2 Diffusion in One Dimension
- 1.3.3 Heat Equation in One Dimension
- 1.3.4 Heat Equation in Three Dimensions
- 1.3.5 String Vibrations and Wave Equation in One Dimension
- 1.3.6 Wave Equation in Two Dimensions – Vibrating Membrane
- 1.3.7 Laplace and Poisson Equations – Steady States
- 1.4 Exercises
- 1.1 Conservation Laws
- 2 Classification, Types of Equations, Boundary and Initial Conditions
- 2.1 Basic Types of Equations
- 2.2 Classical, General, Generic and Particular Solutions
- 2.3 Boundary and Initial Conditions
- 2.4 Well-Posed and Ill-Posed Problems
- 2.5 Classification of Linear Equations of the Second Order
- 2.6 Exercises
- 3 Linear Partial Differential Equations of the First Order
- 3.1 Equations with Constant Coefficients
- 3.1.1 Geometric Interpretation – Method of Characteristics
- 3.1.2 Coordinate Method
- 3.1.3 Method of Characteristic Coordinates
- 3.2 Equations with Non-Constant Coefficients
- 3.2.1 Method of Characteristics
- 3.2.2 Method of Characteristic Coordinates
- 3.3 Problems with Side Conditions
- 3.4 Solution in Parametric Form
- 3.5 Exercises
- 3.1 Equations with Constant Coefficients
- 4 Wave Equation in One Spatial Variable – Cauchy Problem in R
- 4.1 General Solution of the Wave Equation
- 4.1.1 Transformation to System of Two First Order Equations
- 4.1.2 Method of Characteristics
- 4.2 Cauchy Problem on the Real Line
- 4.3 Principle of Causality
- 4.4 Wave Equation with Sources
- 4.4.1 Use of Green’s Theorem
- 4.4.2 Operator Method
- 4.5 Exercises
- 4.1 General Solution of the Wave Equation
- 5 Diffusion Equation in One Spatial Variable – Cauchy Problem in R
- 5.1 Cauchy Problem on the Real Line
- 5.2 Diffusion Equation with Sources
- 5.3 Exercises
- 6 Laplace and Poisson Equations in Two Dimensions
- 6.1 Invariance of the Laplace Operator
- 6.2 Transformation of the Laplace Operator into Polar Coordinates
- 6.3 Solutions of Laplace and Poisson Equations in R2
- 6.3.1 Laplace Equation
- 6.3.2 Poisson Equation
- 6.4 Exercises
- 7 Solutions of Initial Boundary Value Problems for Evolution Equations
- 7.1 Initial Boundary Value Problems on Half-Line
- 7.1.1 Diffusion and Heat Flow on Half-Line
- 7.1.2 Wave on the Half-Line
- 7.1.3 Problems with Nonhomogeneous Boundary Condition
- 7.2 Initial Boundary Value Problem on Finite Interval, Fourier Method
- 7.2.1 Dirichlet Boundary Conditions, Wave Equation
- 7.2.2 Dirichlet Boundary Conditions, Diffusion Equation
- 7.2.3 Neumann Boundary Conditions
- 7.2.4 Robin Boundary Conditions
- 7.2.5 Principle of the Fourier Method
- 7.3 Fourier Method for Nonhomogeneous Problems
- 7.3.1 Nonhomogeneous Equation
- 7.3.2 Nonhomogeneous Boundary Conditions and Their Transformation
- 7.4 Transformation to Simpler Problems
- 7.4.1 Lateral Heat Transfer in Bar
- 7.4.2 Problem with Convective Term
- 7.5 Exercises
- 7.1 Initial Boundary Value Problems on Half-Line
- 8 Solutions of Boundary Value Problems for Stationary Equations
- 8.1 Laplace Equation on Rectangle
- 8.2 Laplace Equation on Disc
- 8.3 Poisson Formula
- 8.4 Exercises
- 9 Methods of Integral Transforms
- 9.1 Laplace Transform
- 9.2 Fourier Transform
- 9.3 Exercises
- 10 General Principles
- 10.1 Principle of Causality (Wave Equation)
- 10.2 Energy Conservation Law (Wave Equation)
- 10.3 Ill-Posed Problem (Diffusion Equation for Negative t)
- 10.4 Maximum Principle (Heat Equation)
- 10.5 Energy Method (Diffusion Equation)
- 10.6 Maximum Principle (Laplace Equation)
- 10.7 Consequences of Poisson Formula (Laplace Equation)
- 10.8 Comparison of Wave, Diffusion and Laplace Equations
- 10.9 Exercises
- 11 Laplace and Poisson equations in Higher Dimensions
- 11.1 Invariance of the Laplace Operator and its Transformation into Spherical Coordinates
- 11.2 Green’s First Identity
- 11.3 Properties of Harmonic Functions
- 11.3.1 Mean Value Property and Strong Maximum Principle
- 11.3.2 Dirichlet Principle
- 11.3.3 Uniqueness of Solution of Dirichlet Problem
- 11.3.4 Necessary Condition for the Solvability of Neumann Problem
- 11.4 Green’s Second Identity and Representation Formula
- 11.5 Boundary Value Problems and Green’s Function
- 11.6 Dirichlet Problem on Half-Space and on Ball
- 11.6.1 Dirichlet Problem on Half-Space
- 11.6.2 Dirichlet Problem on a Ball
- 11.7 Exercises
- 12 Diffusion Equation in Higher Dimensions
- 12.1 Cauchy Problem in R3
- 12.1.1 Homogeneous Problem
- 12.1.2 Nonhomogeneous Problem
- 12.2 Diffusion on Bounded Domains, Fourier Method
- 12.2.1 Fourier Method
- 12.2.2 Nonhomogeneous Problems
- 12.3 General Principles for Diffusion Equation
- 12.4 Exercises
- 12.1 Cauchy Problem in R3
- 13 Wave Equation in Higher Dimensions
- 13.1 Cauchy Problem in R3 – Kirchhoff’s Formula
- 13.2 Cauchy Problem in R2
- 13.3 Wave with Sources in R3
- 13.4 Characteristics, Singularities, Energy and Principle of Causality
- 13.4.1 Characteristics
- 13.4.2 Energy
- 13.4.3 Principle of Causality
- 13.5 Wave on Bounded Domains, Fourier Method
- 13.6 Exercises
- A Sturm-Liouville Problem
- B Bessel Functions
- Some Typical Problems Considered in this Book
- Notation
- Bibliography
- Index
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