Computational Partial Differential Equations Using MATLAB

This is a concise yet solid introduction to advanced numerical methods. 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.It helps students better understand the numerical methods through the use of MATLAB[registered].

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.