The use of difference matrices and high-level MATLAB® commands to implement finite difference algorithms is pedagogically novel. This unique and concise textbook gives the reader easy access and a general ability to use first and second difference matrices to set up and solve linear and nonlinear systems in MATLAB which approximate ordinary and partial differential equations. Prerequisites include a knowledge of basic calculus, linear algebra, and ordinary differential equations. Some knowledge of partial differential equations is a plus though the text may easily serve as a supplement for the student currently working through an introductory PDEs course. Familiarity with MATLAB is not required though a little prior experience with programming would be helpful.
In addition to its special focus on solving in MATLAB, the abundance of examples and exercises make this text versatile in use. It would serve well in a graduate course in introductory scientific computing for partial differential equations. With prerequisites mentioned above plus some elementary numerical analysis, most of the material can be covered and many of the exercises assigned in a single semester course. Some of the more challenging exercises make substantial projects and relate to topics from other typical graduate mathematics courses, e.g., linear algebra, differential equations, or topics in nonlinear functional analysis. A selection of the exercises may be assigned as projects throughout the semester. The student will develop the skills to run simulations corresponding to the primarily theoretical course material covered by the instructor. The book can serve as a supplement for the instructor teaching any course in differential equations. Many of the examples can be easily implemented and the resulting simulation demonstrated by the instructor. If the course has a numerical component, a few of the more difficult exercises may be assigned as student projects. Established researchers in theoretical partial differential equations may find this book useful as well, particularly as an introductory guide for their research students. Those unfamiliar with MATLAB can use the material as a reference to quickly develop their own applications in that language. Practical assistance in implementing algorithms in MATLAB can be found in these pages. A mathematician who is new to the practical implementation of methods for scientific computation in general can learn how to implement and execute numerical simulations of differential equations in MATLAB with relative ease by working through a selection of exercises. Additionally, the book can serve as a practical guide in independent study, undergraduate or graduate research experiences, or for reference in simulating solutions to specific thesis or dissertation-related experiments.
Author(s): John M. Neuberger
Publisher: Springer
Year: 2023
Language: English
Pages: 211
City: Cham
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
