Computational Partial Differential Equations Using MATLAB: (Textbooks in Mathematics)

This textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include the classic finite difference method and the finite element method as well as state-of-the-art numerical methods, such as the high-order compact difference method and the radial basis function meshless method. Helps Students Better Understand Numerical Methods through Use of MATLAB(R) The authors uniquely emphasize both theoretical numerical analysis and practical implementation of the algorithms in MATLAB, making the book useful for students in computational science and engineering. They provide students with simple, clear implementations instead of sophisticated usages of MATLAB functions. All the Material Needed for a Numerical Analysis Course Based on the authors' own courses, the text only requires some knowledge of computer programming, advanced calculus, and difference equations. It includes practical examples, exercises, references, and problems, along with a solutions manual for qualifying instructors. Students can download MATLAB code from www.crcpress.com, enabling them to easily modify or improve the codes to solve their own problems.

Table of Contents:

Brief Overview of Partial Differential Equations
The parabolic equations
The wave equations
The elliptic equations
Differential equations in broader areas
A quick review of numerical methods for PDEs

Finite Difference Methods for Parabolic Equations
Introduction
Theoretical issues: stability, consistence, and convergence
1-D parabolic equations
2-D and 3-D parabolic equations
Numerical examples with MATLAB codes

Finite Difference Methods for Hyperbolic Equations
Introduction
Some basic difference schemes
Dissipation and dispersion errors
Extensions to conservation laws
The second-order hyperbolic PDEs
Numerical examples with MATLAB codes

Finite Difference Methods for Elliptic Equations
Introduction
Numerical solution of linear systems
Error analysis with a maximum principle
Some extensions
Numerical examples with MATLAB codes

High-Order Compact Difference Methods
1-D problems
High-dimensional problems
Other high-order compact schemes

Finite Element Methods: Basic Theory
Introduction to 1-D problems
Introduction to 2-D problems
Abstract finite element theory
Examples of conforming finite element spaces
Examples of nonconforming finite elements
Finite element interpolation theory
Finite element analysis of elliptic problems
Finite element analysis of time-dependent problems

Finite Element Methods: Programming
Finite element method mesh generation
Forming finite element method equations
Calculation of element matrices
Assembly and implementation of boundary conditions
The MATLAB code for P₁ element
The MATLAB code for the Q₁ element

Mixed Finite Element Methods
An abstract formulation
Mixed methods for elliptic problems
Mixed methods for the Stokes problem
An example MATLAB code for the Stokes problem
Mixed methods for viscous incompressible flows

Finite Element Methods for Electromagnetics
Introduction to Maxwell’s equations
The time-domain finite element method
The frequency-domain finite element method
Maxwell’s equations in dispersive media

Meshless Methods with Radial Basis Functions
Introduction
The radial basis functions
The MFS-DRM
Kansa’s method
Numerical examples with MATLAB codes
Coupling RBF meshless methods with DDM

Other Meshless Methods
Construction of meshless shape functions
The element-free Galerkin method
The meshless local Petrov–Galerkin method

Answers to Selected Problems
Index

Bibliographical remarks, Exercises, and References appear at the end of each chapter.