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An Introduction to Partial Differential Equations

Book Description

A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, physics and engineering. The presentation is lively and up to date, paying particular emphasis to developing an appreciation of underlying mathematical theory. Beginning with basic definitions, properties and derivations of some basic equations of mathematical physics from basic principles, the book studies first order equations, classification of second order equations, and the one-dimensional wave equation. Two chapters are devoted to the separation of variables, whilst others concentrate on a wide range of topics including elliptic theory, Green's functions, variational and numerical methods. A rich collection of worked examples and exercises accompany the text, along with a large number of illustrations and graphs to provide insight into the numerical examples. Solutions to selected exercises are included for students whilst extended solution sets are available to lecturers from solutions@cambridge.org.

Note:The ebook version does not provide access to the companion files.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 1. Introduction
    1. 1.1 Preliminaries
    2. 1.2 Classification
    3. 1.3 Differential operators and the superposition principle
    4. 1.4 Differential equations as mathematical models
    5. 1.5 Associated conditions
    6. 1.6 Simple examples
    7. 1.7 Exercises
  9. 2. First-order equations
    1. 2.1 Introduction
    2. 2.2 Quasilinear equations
    3. 2.3 The method of characteristics
    4. 2.4 Examples of the characteristics method
    5. 2.5 The existence and uniqueness theorem
    6. 2.6 The Lagrange method
    7. 2.7 Conservation laws and shock waves
    8. 2.8 The eikonal equation
    9. 2.9 General nonlinear equations
    10. 2.10 Exercises
  10. 3. Second-order linear equations in two independent variables
    1. 3.1 Introduction
    2. 3.2 Classification
    3. 3.3 Canonical form of hyperbolic equations
    4. 3.4 Canonical form of parabolic equations
    5. 3.5 Canonical form of elliptic equations
    6. 3.6 Exercises
  11. 4. The one-dimensional wave equation
    1. 4.1 Introduction
    2. 4.2 Canonical form and general solution
    3. 4.3 The Cauchy problem and d’Alembert’s formula
    4. 4.4 Domain of dependence and region of influence
    5. 4.5 The Cauchy problem for the nonhomogeneous wave equation
    6. 4.6 Exercises
  12. 5. The method of separation of variables
    1. 5.1 Introduction
    2. 5.2 Heat equation: homogeneous boundary condition
    3. 5.3 Separation of variables for the wave equation
    4. 5.4 Separation of variables for nonhomogeneous equations
    5. 5.5 The energy method and uniqueness
    6. 5.6 Further applications of the heat equation
    7. 5.7 Exercises
  13. 6. Sturm–Liouville problems and eigenfunction expansions
    1. 6.1 Introduction
    2. 6.2 The Sturm–Liouville problem
    3. 6.3 Inner product spaces and orthonormal systems
    4. 6.4 The basic properties of Sturm–Liouville eigenfunctions and eigenvalues
    5. 6.5 Nonhomogeneous equations
    6. 6.6 Nonhomogeneous boundary conditions
    7. 6.7 Exercises
  14. 7. Elliptic equations
    1. 7.1 Introduction
    2. 7.2 Basic properties of elliptic problems
    3. 7.3 The maximum principle
    4. 7.4 Applications of the maximum principle
    5. 7.5 Green’s identities
    6. 7.6 The maximum principle for the heat equation
    7. 7.7 Separation of variables for elliptic problems
    8. 7.8 Poisson’s formula
    9. 7.9 Exercises
  15. 8. Green’s functions and integral representations
    1. 8.1 Introduction
    2. 8.2 Green’s function for Dirichlet problem in the plane
    3. 8.3 Neumann’s function in the plane
    4. 8.4 The heat kernel
    5. 8.5 Exercises
  16. 9. Equations in high dimensions
    1. 9.1 Introduction
    2. 9.2 First-order equations
    3. 9.3 Classification of second-order equations
    4. 9.4 The wave equation in R2 and R3
    5. 9.5 The eigenvalue problem for the Laplace equation
    6. 9.6 Separation of variables for the heat equation
    7. 9.7 Separation of variables for the wave equation
    8. 9.8 Separation of variables for the Laplace equation
    9. 9.9 Schrödinger equation for the hydrogen atom
    10. 9.10 Musical instruments
    11. 9.11 Green’s functions in higher dimensions
    12. 9.12 Heat kernel in higher dimensions
    13. 9.13 Exercises
  17. 10. Variational methods
    1. 10.1 Calculus of variations
    2. 10.2 Function spaces and weak formulation
    3. 10.3 Exercises
  18. 11. Numerical methods
    1. 11.1 Introduction
    2. 11.2 Finite differences
    3. 11.3 The heat equation: explicit and implicit schemes,stability, consistency and convergence
    4. 11.4 Laplace equation
    5. 11.5 The wave equation
    6. 11.6 Numerical solutions of large linear algebraic systems
    7. 11.7 The finite elements method
    8. 11.8 Exercises
  19. 12. Solutions of odd-numbered problems
  20. A.1. Trigonometric formulas
  21. A.2. Integration formulas
  22. A.3. Elementary ODEs
  23. A.4. Differential operators in polar coordinates
  24. A.5. Differential operators in spherical coordinates
  25. References
  26. Index