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.
Includes bibliographical references and index.
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