Partial Differential Equations: Analytical and Numerical Methods, Second Edition

Content: 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(& Omega
) --
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.