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| 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. |
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