Computational Partial Differential Equations Using MATLAB - Second Edition
The purpose of this book is to provide a quick but solid introduction to advanced numerical methods for solving various partial differential equations (PDEs) in science and engineering. The numerical methods covered in this book include not only the classic finite difference and finite element methods, but also some recently developed meshless methods, high-order compact difference methods, and finite element methods for Maxwell’s equations in complex media.
This book is based on the material that we have taught in our numerical analysis courses, MAT 665/666 and MAT 765/766, at the University of Nevada Las Vegas since 2003. The emphasis of the text is on both mathematical theory and practical implementation of the numerical methods. We have tried to keep the mathematics accessible for a broad audience, while still presenting the results as rigorously as possible.
This book covers three types of numerical methods for PDEs: the finite difference method, the finite element method, and the meshless method. In Chapter 1, we provide a brief overview of some interesting PDEs coming from different areas and a short review of numerical methods for PDEs. Then we introduce the finite difference methods for solving parabolic, hyperbolic, and elliptic equations in Chapters 2, 3, and 4, respectively. Chapter 5 presents the high-order compact difference method, which is quite popular for solving time-dependent wave propagation problems. Chapters 6 through 9 cover the finite element method. In Chapter 6, fundamental finite element theory is introduced, while in Chapter 7, basic finite element programming techniques are presented. Then in Chapter 8, we extend the discussion to the mixed finite element method. Here both theoretical analysis and programming implementation are introduced. In Chapter 9, we focus on some special finite element methods for solving Maxwell’s equations, where some newly developed algorithms and Maxwell’s equations in dispersive media are presented. Chapter 10 is devoted to the radial basis function meshless methods developed in recent years. Some Galerkin-type meshless methods are introduced in Chapter 11.
The book is intended to be an advanced textbook on numerical methods applied to diverse PDEs such as elliptic, parabolic, and hyperbolic equations. Each chapter includes about 10 exercises for readers to practice and enhance their understanding of the materials. This book supplies many MATLAB source codes, which hopefully will help readers better understand the presented numerical methods. We want to emphasize that our goal is to provide readers with simple and clear implementations instead of sophisticated usages of MATLAB functions. The skilled reader should be able to easily modify or improve the codes to solve similar problems of his or her interest.
This book can be used for a two-semester graduate course that provides an introduction to numerical methods for partial differential equations. The first semester can cover the elementary chapters such as Chapters 1 through 5. This part can also be used at the undergraduate level as a one-semester introductory course on numerical methods or scientific computing. The rest of the chapters are more advanced and can be used for the second semester or a stand-alone advanced numerical analysis course .
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