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Partial Differential Equations
 
Partial Differential Equations
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7505-5
Product Code:  CHEL/325.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-7519-2
Product Code:  CHEL/325.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-1-4704-7505-5
eBook: ISBN:  978-1-4704-7519-2
Product Code:  CHEL/325.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
Partial Differential Equations
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Partial Differential Equations
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7505-5
Product Code:  CHEL/325.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-7519-2
Product Code:  CHEL/325.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-1-4704-7505-5
eBook ISBN:  978-1-4704-7519-2
Product Code:  CHEL/325.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 3251964; 672 pp
    MSC: Primary 35;

    This book is a gem. It fills the gap between the standard introductory material on PDEs that an undergraduate is likely to encounter after a good ODE course (separation of variables, the basics of the second-order equations from mathematical physics) and the advanced methods (such as Sobolev spaces and fixed point theorems) that one finds in modern books. Although this is not designed as a textbook for applied mathematics, the approach is strongly informed by applications. For instance, there are many existence and uniqueness results, but they are usually approached via very concrete techniques.

    The text contains the standard topics that one expects in an intermediate PDE course: the Dirichlet and Neumann problems, Cauchy's problem, characteristics, the fundamental solution, PDEs in the complex domain, plus a chapter on finite differences, on nonlinear fluid mechanics, and another on integral equations. It is an excellent text for advanced undergraduates or beginning graduate students in mathematics or neighboring fields, such as engineering and physics, where PDEs play a central role.

    Readership

    Graduate students, research mathematicians, engineers and physicists working in PDEs.

  • Table of Contents
     
     
    • Front Cover
    • Preface
    • Contents
    • 1 The Method of Power Series
    • 1. INTRODUCTION
    • 2. THE CAUCHY-KOWALEWSKI THEOREM
    • 2 Equations of the First Order
    • 1. CHARACTERISTICS
    • 2. THE MONGE CONE
    • 3. THE COMPLETE INTEGRAL
    • 4. THE EQUATION OF GEOMETRICAL OPTICS
    • 5. THE HAMILTON-JACOBI THEORY
    • 6. APPLICATIONS
    • 3 Classification of Partial Differential Equations
    • 1. REDUCTION OF LINEAR EQUATIONS IN TWO INDEPENDENT VARIABLES TO CANONICAL FORM
    • 2. EQUATIONS IN MORE INDEPENDENT VARIABLES
    • 3. LORENTZ TRANSFORMATIONS AND SPECIAL RELATIVITY
    • 4. QUASI-LINEAR EQUATIONS IN TWO INDEPENDENT VARIABLES
    • 5. HYPERBOLIC SYSTEMS
    • 4 Cauchy's Problem for Equations with Two Independent Variables
    • 1. CHARACTERISTICS
    • 2. THE METHOD OF SUCCESSIVE APPROXIMATIONS
    • 3. HYPERBOLIC SYSTEMS
    • 4. THE RIEMANN FUNCTION
    • 5 The Fundamental Solution
    • 1. TWO INDEPENDENT VARIABLES
    • 2. SEVERAL INDEPENDENT VARIABLES
    • 3. THE PARAMETRIX
    • 6 Cauchy's Problem in Space of Higher Dimension
    • 1. CHARACTERISTICS; UNIQUENESS
    • 2. THE WAVE EQUATION
    • 3. THE METHOD OF DESCENT
    • 4. EQUATIONS WITH ANALYTIC COEFFICIENTS
    • 7 The Dirichlet and Neumann Problems
    • 1. UNIQUENESS
    • 2. THE GREEN'S AND NEUMANN'S FUNCTIONS
    • 3. THE KERNEL FUNCTION; CONFORMAL MAPPING
    • 8 Dirichlet's Principle
    • 1. ORTHOGONAL PROJECTION
    • 2. EXISTENCE PROOF MOTIVATED BY THE KERNEL FUNCTION
    • 3. EXISTENCE PROOF BASED ON DIRICHLET'S PRINCIPLE
    • 9 Existence Theorems of Potential Theory
    • 1. THE HAHN-BANACH THEOREM
    • 2. SUBHARMONIC FUNCTIONS; BARRIERS
    • 3. REDUCTION TO A FREDHOLM INTEGRAL EQUATION
    • 10 Integral Equations
    • 1. THE FREDHOLM ALTERNATIVE
    • 2. APPLICATIONS
    • 3. EIGENVALUES AND EIGENFUNCTIONS OF A SYMMETRIC KERNEL
    • 4. COMPLETENESS OF EIGENFUNCTION EXPANSIONS
    • 11 Eigenvalue Problems
    • 1. THE VIBRATING MEMBRANE; RAYLEIGH'S QUOTIENT
    • 2. ASYMPTOTIC DISTRIBUTION OF EIGENV ALVES
    • 3. UPPER AND LOWER BOUNDS; SYMMETRIZATION
    • 12 Tricomi's Problem; Formulation of Well Posed Problems
    • 1. EQUATIONS OF MIXED TYPE; UNIQUENESS
    • 2. ENERGY INTEGRAL METHOD FOR SYMMETRIC HYPERBOLIC SYSTEMS
    • 3. INCORRECT PROBLEMS; EQUATIONS WITH NO SOLUTION
    • 13 Finite Differences
    • 1. FORMULATION OF DIFFERENCE EQUATIONS
    • 2. HYPERBOLIC EQUATIONS
    • 3. PARABOLIC EQUATIONS
    • 4. ELLIPTIC EQUATIONS
    • 5. RELAXATION AND OTHER ITERATIVE METHODS
    • 6. EXISTENCE THEOREM FOR THE HEAT EQUATION
    • 14 Fluid Dynamics
    • 1. FORMULATION OF THE EQUATIONS OF MOTION
    • 2. ONE-DIMENSIONAL FLOW
    • 3. STEADY SUBSONIC FLOW
    • 4. TRANSONIC AND SUPERSONIC FLOW
    • 5. FLOWS WITH FREE STREAMLINES
    • 6. MAGNETOHYDRODYNAMICS
    • 15 Free Boundary Problems
    • 1. HADAMARD'S VARIATIONAL FORMULA
    • 2. EXISTENCE OF FLOWS WITH FREE STREAMLINES
    • 3. HYDROMAGNETIC STABILITY
    • 4. THE PLATEAU PROBLEM
    • 16 Partial Differential Equations in the Complex Domain
    • 1. CAUCHY'S PROBLEM FOR ANALYTIC SYSTEMS
    • 2. CHARACTERISTIC COORDINATES
    • 3. ANALYTICITY OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS
    • 4. REFLECTION
    • Bibliography
    • Index
    • Back Cover
  • Reviews
     
     
    • From a review of the original edition:

      This book is primarily a text for a graduate course in partial differential equations, although the later chapters are devoted to special topics not ordinarily covered in books in this field ... [T]he author has made use of an interesting combination of classical and modern analysis in his proofs ... Because of the author's emphasis on constructive methods for solving problems which are of physical interest, his book will likely be as welcome to the engineer and the physicist as to the mathematician ... The author and publisher are to be complimented on the general appearance of the book.

      Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 3251964; 672 pp
MSC: Primary 35;

This book is a gem. It fills the gap between the standard introductory material on PDEs that an undergraduate is likely to encounter after a good ODE course (separation of variables, the basics of the second-order equations from mathematical physics) and the advanced methods (such as Sobolev spaces and fixed point theorems) that one finds in modern books. Although this is not designed as a textbook for applied mathematics, the approach is strongly informed by applications. For instance, there are many existence and uniqueness results, but they are usually approached via very concrete techniques.

The text contains the standard topics that one expects in an intermediate PDE course: the Dirichlet and Neumann problems, Cauchy's problem, characteristics, the fundamental solution, PDEs in the complex domain, plus a chapter on finite differences, on nonlinear fluid mechanics, and another on integral equations. It is an excellent text for advanced undergraduates or beginning graduate students in mathematics or neighboring fields, such as engineering and physics, where PDEs play a central role.

Readership

Graduate students, research mathematicians, engineers and physicists working in PDEs.

  • Front Cover
  • Preface
  • Contents
  • 1 The Method of Power Series
  • 1. INTRODUCTION
  • 2. THE CAUCHY-KOWALEWSKI THEOREM
  • 2 Equations of the First Order
  • 1. CHARACTERISTICS
  • 2. THE MONGE CONE
  • 3. THE COMPLETE INTEGRAL
  • 4. THE EQUATION OF GEOMETRICAL OPTICS
  • 5. THE HAMILTON-JACOBI THEORY
  • 6. APPLICATIONS
  • 3 Classification of Partial Differential Equations
  • 1. REDUCTION OF LINEAR EQUATIONS IN TWO INDEPENDENT VARIABLES TO CANONICAL FORM
  • 2. EQUATIONS IN MORE INDEPENDENT VARIABLES
  • 3. LORENTZ TRANSFORMATIONS AND SPECIAL RELATIVITY
  • 4. QUASI-LINEAR EQUATIONS IN TWO INDEPENDENT VARIABLES
  • 5. HYPERBOLIC SYSTEMS
  • 4 Cauchy's Problem for Equations with Two Independent Variables
  • 1. CHARACTERISTICS
  • 2. THE METHOD OF SUCCESSIVE APPROXIMATIONS
  • 3. HYPERBOLIC SYSTEMS
  • 4. THE RIEMANN FUNCTION
  • 5 The Fundamental Solution
  • 1. TWO INDEPENDENT VARIABLES
  • 2. SEVERAL INDEPENDENT VARIABLES
  • 3. THE PARAMETRIX
  • 6 Cauchy's Problem in Space of Higher Dimension
  • 1. CHARACTERISTICS; UNIQUENESS
  • 2. THE WAVE EQUATION
  • 3. THE METHOD OF DESCENT
  • 4. EQUATIONS WITH ANALYTIC COEFFICIENTS
  • 7 The Dirichlet and Neumann Problems
  • 1. UNIQUENESS
  • 2. THE GREEN'S AND NEUMANN'S FUNCTIONS
  • 3. THE KERNEL FUNCTION; CONFORMAL MAPPING
  • 8 Dirichlet's Principle
  • 1. ORTHOGONAL PROJECTION
  • 2. EXISTENCE PROOF MOTIVATED BY THE KERNEL FUNCTION
  • 3. EXISTENCE PROOF BASED ON DIRICHLET'S PRINCIPLE
  • 9 Existence Theorems of Potential Theory
  • 1. THE HAHN-BANACH THEOREM
  • 2. SUBHARMONIC FUNCTIONS; BARRIERS
  • 3. REDUCTION TO A FREDHOLM INTEGRAL EQUATION
  • 10 Integral Equations
  • 1. THE FREDHOLM ALTERNATIVE
  • 2. APPLICATIONS
  • 3. EIGENVALUES AND EIGENFUNCTIONS OF A SYMMETRIC KERNEL
  • 4. COMPLETENESS OF EIGENFUNCTION EXPANSIONS
  • 11 Eigenvalue Problems
  • 1. THE VIBRATING MEMBRANE; RAYLEIGH'S QUOTIENT
  • 2. ASYMPTOTIC DISTRIBUTION OF EIGENV ALVES
  • 3. UPPER AND LOWER BOUNDS; SYMMETRIZATION
  • 12 Tricomi's Problem; Formulation of Well Posed Problems
  • 1. EQUATIONS OF MIXED TYPE; UNIQUENESS
  • 2. ENERGY INTEGRAL METHOD FOR SYMMETRIC HYPERBOLIC SYSTEMS
  • 3. INCORRECT PROBLEMS; EQUATIONS WITH NO SOLUTION
  • 13 Finite Differences
  • 1. FORMULATION OF DIFFERENCE EQUATIONS
  • 2. HYPERBOLIC EQUATIONS
  • 3. PARABOLIC EQUATIONS
  • 4. ELLIPTIC EQUATIONS
  • 5. RELAXATION AND OTHER ITERATIVE METHODS
  • 6. EXISTENCE THEOREM FOR THE HEAT EQUATION
  • 14 Fluid Dynamics
  • 1. FORMULATION OF THE EQUATIONS OF MOTION
  • 2. ONE-DIMENSIONAL FLOW
  • 3. STEADY SUBSONIC FLOW
  • 4. TRANSONIC AND SUPERSONIC FLOW
  • 5. FLOWS WITH FREE STREAMLINES
  • 6. MAGNETOHYDRODYNAMICS
  • 15 Free Boundary Problems
  • 1. HADAMARD'S VARIATIONAL FORMULA
  • 2. EXISTENCE OF FLOWS WITH FREE STREAMLINES
  • 3. HYDROMAGNETIC STABILITY
  • 4. THE PLATEAU PROBLEM
  • 16 Partial Differential Equations in the Complex Domain
  • 1. CAUCHY'S PROBLEM FOR ANALYTIC SYSTEMS
  • 2. CHARACTERISTIC COORDINATES
  • 3. ANALYTICITY OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS
  • 4. REFLECTION
  • Bibliography
  • Index
  • Back Cover
  • From a review of the original edition:

    This book is primarily a text for a graduate course in partial differential equations, although the later chapters are devoted to special topics not ordinarily covered in books in this field ... [T]he author has made use of an interesting combination of classical and modern analysis in his proofs ... Because of the author's emphasis on constructive methods for solving problems which are of physical interest, his book will likely be as welcome to the engineer and the physicist as to the mathematician ... The author and publisher are to be complimented on the general appearance of the book.

    Mathematical Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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