Partial differential equations : analytical and numerical methods / Mark S. Gockenbach.

Gockenbach, Mark S.
Philadelphia : Society for Industrial and Applied Mathematics, c2011.
xx, 654 p. : ill. ; 27 cm.
2nd ed.

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Differential equations, Partial.
Machine generated contents note: 2.1. Heat flow in a bar; Fourier's law
2.1.1. Boundary and initial conditions for the heat equation
2.1.2. Steady-state heat flow
2.1.3. Diffusion
2.2. The hanging bar
2.2.1. Boundary conditions for the hanging bar
2.3. The wave equation for a vibrating string
2.4. Advection; kinematic waves
2.4.1. Initial/boundary conditions for the advection equation
2.4.2. The advection-diffusion equation
2.4.3. Conservation laws
2.4.4. Burgers's equation
2.5. Suggestions for further reading
3.1. Linear systems as linear operator equations
3.2. Existence and uniqueness of solutions to Ax = b
3.2.1. Existence
3.2.2. Uniqueness
3.2.3. The Fredholm alternative
3.3. Basis and dimension
3.4. Orthogonal bases and projections
3.4.1. The L2 inner product
3.4.2. The projection theorem
3.5. Eigenvalues and eigenvectors of a symmetric matrix
3.5.1. The transpose of a matrix and the dot product
3.5.2. Special properties of symmetric matrices
3.5.3. The spectral method for solving Ax = b
3.6. Preview of methods for solving ODEs and PDEs
3.7. Suggestions for further reading
4.1. Background
4.1.1. Converting a higher-order equation to a first-order system
4.1.2. The general solution of a homogeneous linear second-order ODE
4.1.3. The Wronskian test
4.2. Solutions to some simple ODEs
4.2.1. The general solution of a second-order homogeneous ODE with constant coefficients
4.2.2. Variation of parameters
4.2.3. A special inhomogeneous second-order linear ODE
4.2.4. First-order linear ODEs
4.2.5. Euler equations
4.3. Linear systems with constant coefficients
4.3.1. Homogeneous systems
4.3.2. Inhomogeneous systems and variation of parameters
4.3.3. Duhamel's principle
4.4. Numerical methods for initial value problems
4.4.1. Euler's method
4.4.2. Improving on Euler's method: Runge[
]Kutta methods
4.4.3. Numerical methods for systems of ODEs
4.4.4. Automatic step control and Runge[
]Fehlberg methods
4.5. Stiff systems of ODEs
4.5.1. A simple example of a stiff system
4.5.2. The backward Euler method
4.6. Suggestions for further reading
5.1. The analogy between BVPs and linear algebraic systems
5.1.1. A note about direct integration
5.2. Introduction to the spectral method; eigenfunctions
5.2.1. Eigenpairs of [
] cyjd2 under Dirichlet conditions
5.2.2. Representing functions in terms of eigenfunctions
5.2.3. Eigenfunctions under other boundary conditions; other Fourier series
5.3. Solving the BVP using Fourier series
5.3.1. A special case
5.3.2. The general case
5.3.3. Other boundary conditions
5.3.4. Inhomogeneous boundary conditions
5.3.5. Summary
5.4. Finite element methods for BVPs
5.4.1. The principle of virtual work and the weak form of a BVP
5.4.2. The equivalence of the strong and weak forms of the BVP
5.5. The Galerkin method
5.6. Piecewise polynomials and the finite element method
5.6.1. Examples using piecewise linear finite elements
5.6.2. Inhomogeneous Dirichlet conditions
5.7. Suggestions for further reading
6.1. Fourier series methods for the heat equation
6.1.1. The homogeneous heat equation
6.1.2. Nondimensionalization
6.1.3. The inhomogeneous heat equation
6.1.4. Inhomogeneous boundary conditions
6.1.5. Steady-state heat flow and diffusion
6.1.6. Separation of variables
6.2. Pure Neumann conditions and the Fourier cosine series
6.2.1. One end insulated; mixed boundary conditions
6.2.2. Both ends insulated; Neumann boundary conditions
6.2.3. Pure Neumann conditions in a steady-state BVP
6.3. Periodic boundary conditions and the full Fourier series
6.3.1. Eigenpairs of [
]ci.d2 under periodic boundary conditions
6.3.2. Solving the BVP using the full Fourier series
6.3.3. Solving the IBVP using the full Fourier series
6.4. Finite element methods for the heat equation
6.4.1. The method of lines for the heat equation
6.5. Finite elements and Neumann conditions
6.5.1. The weak form of a BVP with Neumann conditions
6.5.2. Equivalence of the strong and weak forms of a BVP with Neumann conditions
6.5.3. Piecewise linear finite elements with Neumann conditions
6.5.4. Inhomogeneous Neumann conditions
6.5.5. The finite element method for an IBVP with Neumann conditions
6.6. Suggestions for further reading
7.1. The homogeneous wave equation without boundaries
7.2. Fourier series methods for the wave equation
7.2.1. Fourier series solutions of the homogeneous wave equation
7.2.2. Fourier series solutions of the inhomogeneous wave equation
7.2.3. Other boundary conditions
7.3. Finite element methods for the wave equation
7.3.1. The wave equation with Dirichlet conditions
7.3.2. The wave equation under other boundary conditions
7.4. Resonance
7.4.1. The wave equation with a periodic boundary condition
7.4.2. The wave equation with a localized source
7.5. Finite difference methods for the wave equation
7.5.1. Finite difference approximation of derivatives
7.5.2. The wave equation
7.5.3. Neumann boundary conditions
7.6. Comparison of the heat and wave equations
7.7. Suggestions for further reading
8.1. The simplest PDE and the method of characteristics
8.1.1. Changing variables
8.1.2. An inhomogeneous PDE
8.2. First-order quasi-linear PDEs
8.2.1. Linear equations
8.2.2. Noncharacteristic initial curves
8.2.3. Semilinear equations
8.2.4. Quasi-linear equations
8.3. Burgers's equation
8.4. Suggestions for further reading
9.1. Green's functions for BVPs in ODEs: Special cases
9.1.1. The Green's function and the inverse of a differential operator
9.1.2. Symmetry of the Green's function; reciprocity
9.2. Green's functions for BVPs in ODEs: The symmetric case
9.2.1. Derivation of the Green's function
9.2.2. Properties of the Green's function; inhomogeneous boundary conditions
9.3. Green's functions for BVPs in ODEs: The general case
9.4. Introduction to Green's functions for IVPs
9.4.1. The Green's function for first-order linear ODEs
9.4.2. The Green's function for higher-order ODEs
9.4.3. Interpretation of the causal Green's function
9.5. Green's functions for the heat equation
9.5.1. The Gaussian kernel
9.5.2. The Green's function on a bounded interval
9.5.3. Properties of the Green's function
9.5.4. Green's functions under other boundary conditions
9.6. Green's functions for the wave equation
9.6.1. The Green's function on the real line
9.6.2. The Green's function on a bounded interval
9.7. Suggestions for further reading
10.1. Introduction
10.1.1. How Sturm[-]Liouville problems arise
10.1.2. Boundary conditions for the Sturm[-]Liouville problem
10.2. Properties of the Sturm[-]Liouville operator
10.2.1. Symmetry
10.2.2. Existence of eigenvalues and eigenfunctions
10.3. Numerical methods for Sturm[-]Liouville problems
10.3.1. The weak form
10.4. Examples of Sturm[-]Liouville problems
10.4.1. A guitar string with variable density
10.4.2. Heat flow with a variable thermal conductivity
10.5. Robin boundary conditions
10.5.1. Eigenvalues under Robin conditions
10.5.2. The nonphysical case
10.6. Finite element methods for Robin boundary conditions
10.6.1. A BVP with a Robin condition
10.6.2. A Sturm[-]Liouville problem with a Robin condition
10.7. The theory of Sturm[-]Liouville problems: An outline
10.7.1. Facts about the eigenvalues
10.7.2. Facts about the eigenfunctions
10.8. Suggestions for further reading
11.1. Physical models in two or three spatial dimensions
11.1.1. The divergence theorem
11.1.2. The heat equation for a three-dimensional domain
11.1.3. Boundary conditions for the three-dimensional heat equation
11.1.4. The heat equation in a bar
11.1.5. The heat equation in two dimensions
11.1.6. The wave equation for a three-dimensional domain
11.1.7. The wave equation in two dimensions
11.1.8. Equilibrium problems and Laplace's equation
11.1.9. Advection and other first-order PDEs
11.1.10. Green's identities and the symmetry of the Laplacian
11.2. Fourier series on a rectangular domain
11.2.1. Dirichlet boundary conditions
11.2.2. Solving a boundary value problem
11.2.3. Time-dependent problems
11.2.4. Other boundary conditions for the rectangle
11.2.5. Neumann boundary conditions
11.2.6. Dirichlet and Neumann problems for Laplace's equation
11.2.7. Fourier series methods for a rectangular box in three dimensions
11.3. Fourier series on a disk
11.3.1. The Laplacian in polar coordinates
11.3.2. Separation of variables in polar coordinates
11.3.3. Bessel's equation
11.3.4. Properties of the Bessel functions
11.3.5. The eigenfunctions of the negative Laplacian on the disk
11.3.6. Solving PDEs on a disk
11.4. Finite elements in two dimensions
11.4.1. The weak form of a BVP in multiple dimensions
11.4.2. Galerkin's method
11.4.3. Piecewise linear finite elements in two dimensions
11.4.4. Finite elements and Neumann conditions
11.4.5. Inhomogeneous boundary conditions
Note continued: 11.5. The free-space Green's function for the Laplacian
11.5.1. The free-space Green's function in two dimensions
11.5.2. The free-space Green's function in three dimensions
11.6. The Green's function for the Laplacian on a bounded domain
11.6.1. Reciprocity
11.6.2. The Green's function for a disk
11.6.3. Inhomogeneous boundary conditions
11.6.4. The Poisson integral formula
11.7. Green's function for the wave equation
11.7.1. The free-space Green's function
11.7.2. The wave equation in two-dimensional space
11.7.3. Huygen's principle
11.7.4. The Green's function for the wave equation on a bounded domain
11.8. Green's functions for the heat equation
11.8.1. The free-space Green's function
11.8.2. The Green's function on a bounded domain
11.9. Suggestions for further reading
12.1. The complex Fourier series
12.1.1. Complex inner products
12.1.2. Orthogonality of the complex exponentials
12.1.3. Representing functions with complex Fourier series
12.1.4. The complex Fourier series of a real-valued function
12.2. Fourier series and the FFT
12.2.1. Using the trapezoidal rule to estimate Fourier coefficients
12.2.2. The discrete Fourier transform
12.2.3. A note about using packaged FFT routines
12.2.4. Fast transforms and other boundary conditions; the discrete sine transform
12.2.5. Computing the DST using the FFT
12.3. Relationship of sine and cosine series to the full Fourier series
12.4. Pointwise convergence of Fourier series
12.4.1. Modes of convergence for sequences of functions
12.4.2. Pointwise convergence of the complex Fourier series
12.5. Uniform convergence of Fourier series
12.5.1. Rate of decay of Fourier coefficients
12.5.2. Uniform convergence
12.5.3. A note about Gibbs's phenomenon
12.6. Mean-square convergence of Fourier series
12.6.1. The space L2([
]l, l)
12.6.2. Mean-square convergence of Fourier series
12.6.3. Cauchy sequences and completeness
12.7. A note about general eigenvalue problems
12.8. Suggestions for further reading
13.1. Implementation of finite element methods
13.1.1. Describing a triangulation
13.1.2. Computing the stiffness matrix
13.1.3. Computing the load vector
13.1.4. Quadrature
13.2. Solving sparse linear systems
13.2.1. Gaussian elimination for dense systems
13.2.2. Direct solution of banded systems
13.2.3. Direct solution of general sparse systems
13.2.4. Iterative solution of sparse linear systems
13.2.5. The conjugate gradient algorithm
13.2.6. Convergence of the CG algorithm
13.2.7. Preconditioned CG
13.3. An outline of the convergence theory for finite element methods
13.3.1. The Sobolev space H01(Ω)
13.3.2. Best approximation in the energy norm
13.3.3. Approximation by piecewise polynomials
13.3.4. Elliptic regularity and L2 estimates
13.4. Finite element methods for eigenvalue problems
13.5. Suggestions for further reading
B.1. Inhomogeneous Dirichlet conditions on a rectangle
B.2. Inhomogeneous Neumann conditions on a rectangle.
Includes bibliographical references and index.
Local notes:
Acquired for the Penn Libraries with assistance from the Class of 1891 Department of Arts Fund.
Class of 1891 Department of Arts Fund.
Publisher Number: