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De Gruyter textbook.
Elements of partial differential equations / Pavel Drábek, Gabriela Holubová. — Second, revised and extended edition. — 1 online resource (291 pages) : illustrations. — (De Gruyter Textbook). — <URL:http://elib.fa.ru/ebsco/809494.pdf>.

Record create date: 2/12/2015

Subject: Differential equations, Partial — Textbooks.; Differential equations, Partial.; MATHEMATICS / Calculus; MATHEMATICS / Mathematical Analysis

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Table of Contents

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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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