Difference Matrices for ODE and PDE. A MATLAB Companion

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Preface Acknowledgements Contents Acronyms 1 Introduction 1.1 A Summary of the Differential Equations We Will Consider 1.2 The Use of MATLAB® and the Student Exercises 1.2.1 Using MATLAB®'s Debugger 1.2.2 Line Numbering in MATLAB® Examples 1.2.3 Reproducing Codes and Exercises 1.2.4 Guidelines to the Homework Exercises 1.3 The Organization of This Text 2 Review of Elementary Numerical Methods and MATLAB® 2.1 Introduction to Basic MATLAB® at the Command Line 2.1.1 MATLAB®: sum, prod, max, min, abs, norm, linspace, for loop, eigs, and sort 2.2 Runge–Kutta Method for Initial Value Problems 2.2.1 The Shooting Method for ODE BVP—an IVP Approach 2.2.2 Comparison of Approximate Solutions to Exact Solutions 2.2.3 MATLAB®: Ones, Zeros, the `:' Iterator, the @ Syntax for Defining Inline Functions, Subfunctions 2.2.4 Rows Versus Columns Part I, Diagnosing Dimension and Size Errors 2.3 Numerical Differentiation and Integration 2.3.1 Higher Order Differences and Symbolic Computation 2.3.2 MATLAB®: kron, spdiags, `backslash' (mldivide), tic, and toc 2.4 Newton's Method for Vector Fields 2.4.1 MATLAB®: if, else, while, fprintf, meshgrid, surf, reshape, find, single indexing, and rows versus columns Part II 2.5 Cubic Spline Solver 2.5.1 Making Animations 2.6 Theory: ODE, Systems, Newton's Method, IVP Solvers … 2.6.1 Some ODE Theory and Techniques 2.6.2 Convergence and Order of Newton's Method 2.6.3 First-Order IVP Numerical Solvers: Euler's and Runge–Kutta's 2.6.4 Difference Formulas and Orders of Approximation Exercises 3 Ordinary Differential Equations 3.1 Second-Order Semilinear Elliptic Boundary Value Problems Exercises 3.2 Linear Ordinary Second-Order BVP Exercises 3.3 Eigenvalues of -D2 and Fourier Series Exercises 3.4 Enforcing Zero Dirichlet, Zero Neumann, and Periodic Boundary Conditions … Exercises 3.5 First-Order Linear Solvers Exercises 3.6 Systems of First-Order Linear Equations for Second-Order IVP Exercises 3.7 First-Order Nonlinear IVP Exercises 3.8 A Practical Guide to Fourier Series 4 Partial Differential Equations 4.1 The Laplacian on the Unit Square 4.2 Creating the Sparse Laplacian Matrix D2 and Eigenvalues Exercises 4.3 Semilinear Elliptic BVP on the Square Exercises 4.4 Laplace's Equation on the Square Exercises 4.5 The Heat Equation 4.5.1 Explicit Method 4.5.2 Implicit Method 4.5.3 Explicit–Implicit Method 4.5.4 The Method of Lines 4.5.5 Fourier Expansion with Numerical Integration 4.5.6 Block Matrix Systems Exercises 4.6 The Wave Equation 4.6.1 The Method of Lines 4.6.2 A Good Explicit Method 4.6.3 Block Matrix Systems and D'Alembert Matrices Exercises 4.7 Tricomi's Equation Exercises 4.8 General Regions 4.8.1 The Laplacian on the Cube 4.8.2 The Laplacian on the Disk 4.8.3 Accurate Eigenvalues of the Laplacian on Disk, Annulus, and Sections 4.8.4 The Laplace–Beltrami Operator on a Spherical Section 4.8.5 A General Region Code Exercises 4.9 First-Order PDE and the Method of Characteristics Exercises 4.10 Theory: Separation of Variables for PDE on Rectangular and Polar Regions 4.10.1 Eigenfunctions of the Laplacian 4.10.2 Laplace's Equation 4.10.3 The Heat Equation 4.10.4 The Wave Equation 5 Advanced Topics in Semilinear Elliptic BVP 5.1 Branch Following and Bifurcation Detection 5.1.1 The Tangent Newton Method for Branch Following 5.1.2 The Secant Method for Bifurcation Detection 5.1.3 Secondary Bifurcations and Branch Switching Exercises 5.2 Mountain Pass and Modified Mountain Pass Algorithms for Semilinear BVP 5.2.1 The MPA 5.2.2 The MMPA Exercises 5.3 The p-Laplacian Exercises Appendix References