skip to content
Partial differential equations : analytical and numerical methods Preview this item
ClosePreview this item

Partial differential equations : analytical and numerical methods

Author: Mark S Gockenbach
Publisher: Philadelphia : Society for Industrial and Applied Mathematics, ©2011.
Edition/Format:   Book : English : 2nd edView all editions and formats
Database:WorldCat
Summary:

A fresh, forward-looking undergraduate textbook that treats the finite element method and classical Fourier series method with equal emphasis.

You are not connected to the University of Washington Libraries network. Access to online content and services may require you to authenticate with your library. OffCampus Access (log in)
Getting this item's online copy... Getting this item's online copy...

Find a copy in the library

Getting this item's location and availability... Getting this item's location and availability...

WorldCat

Find it in libraries globally
Worldwide libraries own this item

Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Mark S Gockenbach
ISBN: 9780898719352 0898719356
OCLC Number: 662152779
Description: xx, 654 p. : ill. ; 27 cm.
Contents: 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[ --
]Kutta[ --
]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.
Responsibility: Mark S. Gockenbach.
Retrieving notes about this item Retrieving notes about this item

Reviews

Editorial reviews

Publisher Synopsis

'I love this book and look forward to using it as a text in the future ... It's the first truly modern approach that I've seen in a PDE text.' Maeve McCarthy, MAA Online

 
User-contributed reviews
Retrieving GoodReads reviews...

Tags

Be the first.

Similar Items

Related Subjects:(1)

User lists with this item (1)

Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

Close Window

Please sign in to WorldCat 

Don't have an account? You can easily create a free account.