Introduction to Applied Partial Differential Equations

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  • Format: Hardcover
  • Copyright: 2012-01-06
  • Publisher: W. H. Freeman

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Drawing on his decade of experience teaching the differential equations course ,John Davis offers a refreshing and effective new approach to partial differential equations that is equal parts computational proficiency, visualization, and physical interpretation of the problem at hand.

Table of Contents

1 Introduction to PDEs
1.1 ODEs vs. PDEs
1.2 How PDEs Are Born: Conservation Laws, Fluids, and Waves
1.3 Boundary Conditions in One Space Dimension
1.4 ODE Solution Methods
2 Fourier's Method: Separation of Variables
2.1 Linear Algebra Concepts
2.2 The General Solution via Eigenfunctions
2.3 The Coefficients via Orthogonality
2.4 Consequences of Orthogonality
2.5 Robin Boundary Conditions
2.6 Nonzero Boundary Conditions: Steady-States and Transients*
3 Fourier Series Theory
3.1 Fourier Series: Sine, Cosine, and Full
3.2 Fourier Series vs. Taylor Series: Global vs. Local Approximations*
3.3 Error Analysis and Modes of Convergence
3.4 Convergence Theorems
3.5 Basic L2 Theory
3.6 The Gibbs Phenomenon*
4 General Orthogonal Series Expansions
4.1 Regular and Periodic Sturm-Liouville Theory
4.2 Singular Sturm-Liouville Theory
4.3 Orthogonal Expansions: Special Functions
4.4 Computing Bessel Functions: The Method of Frobenius
4.5 The Gram-Schmidt Procedure*
5 PDEs in Higher Dimensions
5.1 Nuggets from Vector Calculus
5.2 Deriving PDEs in Higher Dimensions
5.3 Boundary Conditions in Higher Dimensions
5.4 Well-Posed Problems: Good Models
5.5 Laplace's Equation in 2D
5.6 The 2D Heat and Wave Equations

6 PDEs in Other Coordinate Systems
6.1 Laplace's Equation in Polar Coordinates
6.2 Poisson's Formula and Its Consequences*
6.3 The Wave Equation and Heat Equation in Polar Coordinates
6.4 Laplace's Equation in Cylindrical Coordinates
6.5 Laplace's Equation in Spherical Coordinates

7 PDEs on Unbounded Domains
7.1 The Infinite String: d'Alembert's Solution
7.2 Characteristic Lines
7.3 The Semi-infinite String: The Method of Reflections
7.4 The Infinite Rod: The Method of Fourier Transforms
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