Computational Partial Differential Equations Using MATLAB

ISBN-10: 1420089048
ISBN-13: 9781420089042
Edition: 2008
List price: $115.95 Buy it from $108.00
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Description: This textbook introduces several major numerical methods for solving partial differential equations. It presents new techniques, such as the high-order compact difference method and the radial basis function meshless method, as well as traditional  More...

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Book details

List price: $115.95
Copyright year: 2008
Publisher: CRC Press LLC
Publication date: 10/20/2008
Binding: Hardcover
Pages: 378
Size: 6.50" wide x 9.50" long x 1.00" tall
Weight: 1.650
Language: English

This textbook introduces several major numerical methods for solving partial differential equations. It presents new techniques, such as the high-order compact difference method and the radial basis function meshless method, as well as traditional techniques that include the classic finite difference method and the finite element method. Ideal for a one- or two-semester course, the text only requires some knowledge of computer programming, advanced calculus, and difference equations. It provides practical examples, exercises, references, and problems, along with a solutions manual for qualifying instructors. An accompanying CD-ROM contains MATLAB source code.

Preface
Acknowledgments
Brief Overview of Partial Differential Equations
The parabolic equations
The wave equations
The elliptic equations
Differential equations in broader areas
Electromagnetics
Fluid mechanics
Ground water contamination
Petroleum reservoir simulation
Finance modeling
Image processing
A quick review of numerical methods for PDEs
References
Finite Difference Methods for Parabolic Equations
Introduction
Theoretical issues: stability, consistence, and convergence
1-D parabolic equations
The [theta]-method
Some extensions
2-D and 3-D parabolic equations
Standard explicit and implicit methods
The ADI methods for 2-D problems
The ADI methods for 3-D problems
Numerical examples with MATLAB codes
Bibliographical remarks
Exercises
References
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
Bibliographical remarks
Exercises
References
Finite Difference Methods for Elliptic Equations
Introduction
Numerical solution of linear systems
Direct methods
Simple iterative methods
Modern iterative methods
Error analysis with a maximum principle
Some extensions
Mixed boundary conditions
Self-adjoint problems
A fourth-order scheme
Numerical examples with MATLAB codes
Bibliographical remarks
Exercises
References
High-Order Compact Difference Methods
One-dimensional problems
Spatial discretization
Approximations of high-order derivatives
Temporal discretization
Low-pass spatial filter
Numerical examples with MATLAB codes
High-dimensional problems
Temporal discretization for 2-D problems
Stability analysis
Extensions to 3-D compact ADI schemes
Numerical examples with MATLAB codes
Other high-order compact schemes
One-dimensional problems
Two-dimensional problems
Bibliographical remarks
Exercises
References
Finite Element Methods: Basic Theory
Introduction to one-dimensional problems
The second-order equation
The fourth-order equation
Introduction to two-dimensional problems
The Poisson's equation
The biharmonic problem
Abstract finite element theory
Existence and uniqueness
Stability and convergence
Examples of conforming finite element spaces
Triangular finite elements
Rectangular finite elements
Examples of nonconforming finite elements
Nonconforming triangular elements
Nonconforming rectangular elements
Finite element interpolation theory
Sobolev spaces
Interpolation theory
Finite element analysis of elliptic problems
Analysis of conforming finite elements
Analysis of nonconforming finite elements
Finite element analysis of time-dependent problems
Introduction
FEM for parabolic equations
Bibliographical remarks
Exercises
References
Finite Element Methods: Programming
FEM mesh generation
Forming FEM equations
Calculation of element matrices
Assembly and implementation of boundary conditions
The MATLAB code for P[subscript 1] element
The MATLAB code for the Q[subscript 1] element
Bibliographical remarks
Exercises
References
Mixed Finite Element Methods
An abstract formulation
Mixed methods for elliptic problems
The mixed variational formulation
The mixed finite element spaces
The error estimates
Mixed methods for the Stokes problem
The mixed variational formulation
Mixed finite element spaces
An example MATLAB code for the Stokes problem
Mixed methods for viscous incompressible flows
The steady Navier-Stokes problem
The unsteady Navier-Stokes problem
Bibliographical remarks
Exercises
References
Finite Element Methods for Electromagnetics
Introduction to Maxwell's equations
The time-domain finite element method
The mixed method
The standard Galerkin method
The discontinuous Galerkin method
The frequency-domain finite element method
The standard Galerkin method
The discontinuous Galerkin method
The mixed DG method
The Maxwell's equations in dispersive media
Isotropic cold plasma
Debye medium
Lorentz medium
Double-negative metamaterials
Bibliographical remarks
Exercises
References
Meshless Methods with Radial Basis Functions
Introduction
The radial basis functions
The MFS-DRM
The fundamental solution of PDEs
The MFS for Laplace's equation
The MFS-DRM for elliptic equations
Computing particular solutions using RBFs
The RBF-MFS
The MFS-DRM for the parabolic equations
Kansa's method
Kansa's method for elliptic problems
Kansa's method for parabolic equations
The Hermite-Birkhoff collocation method
Numerical examples with MATLAB codes
Elliptic problems
Biharmonic problems
Coupling RBF meshless methods with DDM
Overlapping DDM
Non-overlapping DDM
One numerical example
Bibliographical remarks
Exercises
References
Other Meshless Methods
Construction of meshless shape functions
The smooth particle hydrodynamics method
The moving least-square approximation
The partition of unity method
The element-free Galerkin method
The meshless local Petrov-Galerkin method
Bibliographical remarks
Exercises
References
Answers to Selected Problems
Index

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