Beginning partial differential equations /

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"Featuring a challenging, yet accessible, introduction to partial differential equations, Beginning Partial Differential Equations provides a solid introduction to partial differential equations, particularly methods of solution based on characteristics, separation of variables, as well as Four...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: O'Neil, Peter V.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Hoboken : Wiley, 2014.
Edición:Third edition.
Temas:
Differential equations, Partial.
Équations aux dérivées partielles.
MATHEMATICS > Differential Equations.
Differential equations, Partial
Acceso en línea:Texto completo (Requiere registro previo con correo institucional)
Tabla de Contenidos:
  • 1. First Ideas
  • 2. Solutions of the Heat Equation
  • 3. Solutions of the Wave Equation
  • 4. Dirichlet and Neumann Problems
  • 4. 7 Existence Theorem for a Dirichlet Problem
  • 5. Fourier Integral Methods of Solution
  • 6. Solutions Using Eigenfunction Expansions
  • 7. Integral Transform Methods of Solution
  • 8. First-Order Equations
  • 9. End Materials.
  • 1. First Ideas
  • 1.1. Two Partial Differential Equations
  • 1.1.1. The Heat, or Diffusion, Equati
  • 1.1.2. The Wave Equation
  • 1.2. Fourier Series
  • 1.2.1. The Fourier Series of a Function
  • 1.2.2. Fourier Sine and Cosine Series
  • 1.3. Two Eigenvalue Problems
  • 1.4. A Proof of the Fourier Convergence Theorem
  • 1.4.1. The Role of Periodicity
  • 1.4.2. Dirichlet's Formula
  • 1.4.3. The Riemann-Lebesgue Lemma
  • 1.4.4. Proof of the Convergence Theorem
  • 2. Solutions of the Heat Equation
  • 2.1. Solutions on an Interval [0, L]
  • 2.1.1. Ends Kept at Temperature Zero
  • 2.1.2. Insulated Ends
  • 2.1.3. Ends at Different Temperatures
  • 2.1.4. A Diffusion Equation with Additional Terms
  • 2.1.5. One Radiating End
  • 2.2. A Nonhomogeneous Problem
  • 2.3. The Heat Equation in Two Space Variables
  • 2.4. The Weak Maximum Principle
  • 3. Solutions of the Wave Equation
  • 3.1. Solutions on Bounded Intervals
  • 3.1.1. Fixed Ends
  • 3.1.2. Fixed Ends with a Forcing Term
  • 3.1.3. Damped Wave Motion
  • 3.2. The Cauchy Problem
  • 3.2.1. d'Alembert's Solution
  • 3.2.1.1. Forward and Backward Waves
  • 3.2.2. The Cauchy Problem on a Half Line
  • 3.2.3. Characteristic Triangles and Quadrilaterals
  • 3.2.4. A Cauchy Problem with a Forcing Term
  • 3.2.5. String with Moving Ends
  • 3.3. The Wave Equation in Higher Dimensions
  • 3.3.1. Vibrations in a Membrane with Fixed Frame
  • 3.3.2. The Poisson Integral Solution
  • 3.3.3. Hadamard's Method of Descent
  • 4. Dirichlet and Neumann Problems
  • 4.1. Laplace's Equation and Harmonic Functions
  • 4.1.1. Laplace's Equation in Polar Coordinates
  • 4.1.2. Laplace's Equation in Three Dimensions
  • 4.2. The Dirichlet Problem for a Rectangle
  • 4.3. The Dirichlet Problem for a Disk
  • 4.3.1. Poisson's Integral Solution
  • 4.4. Properties of Harmonic Functions
  • 4.4.1. Topology of Rn
  • 4.4.2. Representation Theorems
  • 4.4.2.1. A Representation Theorem in R3
  • 4.4.2.2. A Representation Theorem in the Plane
  • 4.4.3. The Mean Value Property and the Maximum Principle
  • 4.5. The Neumann Problem
  • 4.5.1. Existence and Uniqueness
  • 4.5.2. Neumann Problem for a Rectangle
  • 4.5.3. Neumann Problem for a Disk
  • 4.6. Poisson's Equation
  • 4. 7 Existence Theorem for a Dirichlet Problem
  • 5. Fourier Integral Methods of Solution
  • 5.1. The Fourier Integral of a Function
  • 5.1.1. Fourier Cosine and Sine Integrals
  • 5.2. The Heat Equation on the Real Line
  • 5.2.1. A Reformulation of the Integral Solution
  • 5.2.2. The Heat Equation on a Half Line
  • 5.3. The Debate over the Age of the Earth
  • 5.4. Burger's Equation
  • 5.4.1. Traveling Wave Solutions of Burger's Equation
  • 5.5. The Cauchy Problem for the Wave Equation
  • 5.6. Laplace's Equation on Unbounded Domains
  • 5.6.1. Dirichlet Problem for the Upper Half Plane
  • 5.6.2. Dirichlet Problem for the Right Quarter Plane
  • 5.6.3. A Neumann Problem for the Upper Half Plane
  • 6. Solutions Using Eigenfunction Expansions
  • 6.1. A Theory of Eigenfunction Expansions
  • 6.1.1. A Closer Look at Expansion Coefficients
  • 6.2. Bessel Functions
  • 6.2.1. Variations on Bessel's Equation
  • 6.2.2. Recurrence Relations
  • 6.2.3. Zeros of Bessel Functions
  • 6.2.4. Fourier-Bessel Expansions
  • 6.3. Applications of Bessel Functions
  • 6.3.1. Temperature Distribution in a Solid Cylinder
  • 6.3.2. Vibrations of a Circular Drum
  • 6.3.3. Oscillations of a Hanging Chain
  • 6.3.4. Did Poe Get His Pendulum Right?
  • 6.4. Legendre Polynomials and Applications
  • 6.4.1. A Generating Function
  • 6.4.2. A Recurrence Relation
  • 6.4.3. Fourier-Legendre Expansions
  • 6.4.4. Zeros of Legendre Polynomials
  • 6.4.5. Steady-State Temperature in a Solid Sphere
  • 6.4.6. Spherical Harmonics
  • 7. Integral Transform Methods of Solution
  • 7.1. The Fourier Transform
  • 7.1.1. Convolution
  • 7.1.2. Fourier Sine and Cosine Transforms
  • 7.2. Heat and Wave Equations
  • 7.2.1. The Heat Equation on the Real Line
  • 7.2.2. Solution by Convolution
  • 7.2.3. The Heat Equation on a Half Line
  • 7.2.4. The Wave Equation by Fourier Transform
  • 7.3. The Telegraph Equation
  • 7.4. The Laplace Transform
  • 7.4.1. Temperature Distribution in a Semi-Infinite Bar
  • 7.4.2. A Diffusion Problem in a Semi-Infinite Medium
  • 7.4.3. Vibrations in an Elastic Bar
  • 8. First-Order Equations
  • 8.1. Linear First-Order Equations
  • 8.2. The Significance of Characteristics
  • 8.3. The Quasi-Linear Equation
  • 9. End Materials
  • 9.1. Notation
  • 9.2. Use of MAPLE
  • 9.2.1. Numerical Computations and Graphing
  • 9.2.2. Ordinary Differential Equations
  • 9.2.3. Integral Transforms
  • 9.2.4. Special Functions
  • 9.3. Answers to Selected Problems.

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