Elements of Partial Differential Equations

Pavel Drábek, Gabriela Holubová

• Publisher: De Gruyter
• Year: 2008
• Print ISBN: 9783110191240
• eBook ISBN: 9783110206753
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This textbook presents a first introduction to PDEs on an elementary level, enabling the reader to understand what partial differential equations are, where they come from and how they can be solved. The intention is that the reader understands the basic principles which are valid for particular types of PDEs, and to acquire some classical methods to solve them, thus the authors restrict their considerations to fundamental types of equations and basic methods. Only basic facts from calculus and linear ordinary differential equations of first and second order are needed as a prerequisite.

• An elementary introduction to the basic principles of partial differential equations.
• With many illustrations.
• The book is addressed to students who intend to specialize in mathematics as well as to students of physics, engineering, and economics.

Table of contents

• Preface
• 1 Mathematical Models, Conservation and Constitutive Laws
• 1.1 Basic Notions
• 1.2 Evolution Conservation Law
• 1.3 Stationary Conservation Law
• 1.4 Conservation Law in One Dimension
• 1.5 Constitutive Laws
• 1.6 Exercises
• 2 Classification, Types of Equations, Boundary and Initial Conditions
• 2.1 Basic Types of Equations, Boundary and Initial Conditions
• 2.2 Classification of Linear Equations of the Second Order
• 2.3 Exercises
• 3 Linear Partial Differential Equations of the First Order
• 3.1 Convection and Transport Equation
• 3.2 Equations with Constant Coefficients
• 3.3 Equations with Non-Constant Coefficients
• 3.4 Exercises
• 4 Wave Equation in One Spatial Variable–Cauchy Problem in ℝ
• 4.1 String Vibrations and Wave Equation in One Dimension
• 4.2 Cauchy Problem on the Real Line
• 4.3 Wave Equation with Sources
• 4.4 Exercises
• 5 Diffusion Equation in One Spatial Variable–Cauchy Problem in ℝ
• 5.1 Diffusion and Heat Equations in One Dimension
• 5.2 Cauchy Problem on the Real Line
• 5.3 Diffusion Equation with Sources
• 5.4 Exercises
• 6 Laplace and Poisson Equations in Two Dimensions
• 6.1 Steady States and Laplace and Poisson Equations
• 6.2 Invariance of the Laplace Operator, Its Transformation into Polar Coordinates
• 6.3 Solution of Laplace and Poisson Equations in ℝ2
• 6.4 Exercises
• 7 Solutions of Initial Boundary Value Problems for Evolution Equations
• 7.1 Initial Boundary Value Problems on Half-Line
• 7.2 Initial Boundary Value Problem on Finite Interval, Fourier Method
• 7.3 Fourier Method for Nonhomogeneous Problems
• 7.4 Transformation to Simpler Problems
• 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
• 11.2 Green’s First Identity
• 11.3 Properties of Harmonic Functions
• 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
• 11.7 Exercises
• 12 Diffusion Equation in Higher Dimensions
• 12.1 Heat Equation in Three Dimensions
• 12.2 Cauchy Problem in ℝ3
• 12.3 Diffusion on Bounded Domains, Fourier Method
• 12.4 Exercises
• 13 Wave Equation in Higher Dimensions
• 13.1 Membrane Vibrations and Wave Equation in Two Dimensions
• 13.2 Cauchy Problem in ℝ3–Kirchhoff’s Formula
• 13.3 Cauchy problem in ℝ2
• 13.4 Wave with sources in ℝ3
• 13.5 Characteristics, Singularities, Energy and Principle of Causality
• 13.6 Wave on Bounded Domains, Fourier Method
• 13.7 Exercises
• 14 Appendix
• 14.1 Sturm-Liouville problem
• 14.2 Bessel Functions
• Some Typical Problems Considered in This Book
• Notation
• Bibliography
• Index