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Beginning Partial Differential Equations (3E) by Peter V. O'Neil

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Beginning Partial Differential Equations (3E) written by Peter V. O'Neil .
This edition is based on four themes: methods of solution of initial-boundary value problems, properties and existence of solutions, applications of partial differential equations, and use of software to carry out computations and graphics.
The focus is on equations of diffusion processes and wave motion, and on Dirichlet and Neumann problems. Following an introductory chapter, we look at methods applied to these equations in bounded and unbounded media, and in one and several space dimensions. The topics are organized to make it easy to match problems in specific settings to methods for writing solutions. Methods include Fourier series and integrals, the use of characteristics, integral solutions, integral transforms, and special functions and eigenfunction expansions. Properties of solutions that are considered include existence and uniqueness issues, maximum and mean value principles, integral representations, and sensitivity of solutions to initial and boundary conditions.
In addition to standard material for an introductory course, topics include traveling-wave solutions of Burger
s equation, damped wave motion, heat and wave equations with forcing terms, a general treatment of eigenfunction expansions, a complete solution of the telegraph equation using the Fourier transform, the use of characteristics to solve Cauchy problems and vibrating string problems with moving ends, double Fourier series solutions, and the PoissonKirchhoff integral solution of the wave equation in two dimensions. There are also proofs of important theorems, including an existence theorem for a Dirichlet problem and a convergence theorem for Fourier series.
Finally, there is a section on the use of MAPLE™ to carry out computations and experiment with graphics. MATLAB @, MATHEMATICA ®, and other packages may also be used for these numerical aspects of partial differential equations.

Title: Beginning Partial Differential Equations
Edition: 3rd
Author(s): Peter V. O'Neil
Publisher: Wiley
Year: 2014
Pages: 453
Type: PDF
Language: English
ISBN: 1118629949,9781118629949
Country: UK
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About Author :-
Peter V. O’Neilis from University of Alabama, at Birmingham, UK.

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Beginning Partial Differential Equations (3E) written by Peter V. O'Neil cover the following topics. 1. First Ideas
1.1 Two Partial Differential Equations
1.1.1 The Heat, or Diffusion, Equation
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
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 Alembert's Solution 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 A Representation Theorem in R3 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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