Taking a practical approach to the subject, Advanced Engineering Mathematics with MATLAB(R), Third Edition continues to integrate technology into the conventional topics of engineering mathematics. The author employs MATLAB to reinforce concepts and solve problems that require heavy computation. MATLAB scripts are available for download at www.crcpress.com Along with new examples, problems, and projects, this updated and expanded edition incorporates several significant improvements.
New to the Third Edition New chapter on Green's functions New section that uses the matrix exponential to solve systems of differential equations More numerical methods for solving differential equations, including Adams--Bashforth and finite element methods New chapter on probability that presents basic concepts, such as mean, variance, and probability density functions New chapter on random processes that focuses on noise and other random fluctuations Suitable for a differential equations course or a variety of engineering mathematics courses, the text covers fundamental techniques and concepts as well as Laplace transforms, separation of variable solutions to partial differential equations, the z-transform, the Hilbert transform, vector calculus, and linear algebra. It also highlights many modern applications in engineering to show how these topics are used in practice. A solutions manual is available for qualifying instructors.
Table of Contents:
COMPLEX VARIABLES
Complex Numbers
Finding Roots
The Derivative in the Complex Plane: The Cauchy–Riemann Equations
Line Integrals
Cauchy–Goursat Theorem
Cauchy’s Integral Formula
Taylor and Laurent Expansions and Singularities
Theory of Residues
Evaluation of Real Definite Integrals
Cauchy’s Principal Value Integral
FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Classification of Differential Equations
Separation of Variables
Homogeneous Equations
Exact Equations
Linear Equations
Graphical Solutions
Numerical Methods
HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Homogeneous Linear Equations with Constant Coefficients
Simple Harmonic Motion
Damped Harmonic Motion
Method of Undetermined Coefficients
Forced Harmonic Motion
Variation of Parameters
Euler–Cauchy Equation
Phase Diagrams
Numerical Methods
FOURIER SERIES
Fourier Series
Properties of Fourier Series
Half-Range Expansions
Fourier Series with Phase Angles
Complex Fourier Series
The Use of Fourier Series in the Solution of Ordinary Differential Equations
Finite Fourier Series
THE FOURIER TRANSFORM
Fourier Transforms
Fourier Transforms Containing the Delta Function
Properties of Fourier Transforms
Inversion of Fourier Transforms
Convolution
Solution of Ordinary Differential Equations by Fourier Transforms
THE LAPLACE TRANSFORM
Definition and Elementary Properties
The Heaviside Step and Dirac Delta Functions
Some Useful Theorems
The Laplace Transform of a Periodic Function
Inversion by Partial Fractions: Heaviside’s Expansion Theorem
Convolution
Integral Equations
Solution of Linear Differential Equations with Constant Coefficients
Inversion by Contour Integration
THE Z-TRANSFORM
The Relationship of the Z-Transform to the Laplace Transform
Some Useful Properties
Inverse Z-Transforms
Solution of Difference Equations
Stability of Discrete-Time Systems
THE HILBERT TRANSFORM
Definition
Some Useful Properties
Analytic Signals
Causality: The Kramers–Kronig Relationship
THE STURM–LIOUVILLE PROBLEM
Eigenvalues and Eigenfunctions
Orthogonality of Eigenfunctions
Expansion in Series of Eigenfunctions
A Singular Sturm–Liouville Problem: Legendre’s Equation
Another Singular Sturm–Liouville Problem: Bessel’s Equation
Finite Element Method
THE WAVE EQUATION
The Vibrating String
Initial Conditions: Cauchy Problem
Separation of Variables
D’Alembert’s Formula
The Laplace Transform Method
Numerical Solution of the Wave Equation
THE HEAT EQUATION
Derivation of the Heat Equation
Initial and Boundary Conditions
Separation of Variables
The Laplace Transform Method
The Fourier Transform Method
The Superposition Integral
Numerical Solution of the Heat Equation
LAPLACE’S EQUATION
Derivation of Laplace’s Equation
Boundary Conditions
Separation of Variables
The Solution of Laplace’s Equation on the Upper Half-Plane
Poisson’s Equation on a Rectangle
The Laplace Transform Method
Numerical Solution of Laplace’s Equation
Finite Element Solution of Laplace’s Equation
GREEN’S FUNCTIONS
What Is a Green’s Function?
Ordinary Differential Equations
Joint Transform Method
Wave Equation
Heat Equation
Helmholtz’s Equation
VECTOR CALCULUS
Review
Divergence and Curl
Line Integrals
The Potential Function
Surface Integrals
Green’s Lemma
Stokes’ Theorem
Divergence Theorem
LINEAR ALGEBRA
Fundamentals of Linear Algebra
Determinants
Cramer’s Rule
Row Echelon Form and Gaussian Elimination
Eigenvalues and Eigenvectors
Systems of Linear Differential Equations
Matrix Exponential
PROBABILITY
Review of Set Theory
Classic Probability
Discrete Random Variables
Continuous Random Variables
Mean and Variance
Some Commonly Used Distributions
Joint Distributions
RANDOM PROCESSES
Fundamental Concepts
Power Spectrum
Differential Equations Forced by Random Forcing
Two-State Markov Chains
Birth and Death Processes
Poisson Processes
Random Walk
ANSWERS TO THE ODD-NUMBERED PROBLEMS
INDEX
About the Author :
Dean G. Duffy is a former instructor at the US Naval Academy and US Military Academy. From 1980 to 2005, he worked on numerical weather prediction, oceanic wave modeling, and dynamical meteorology at NASA’s Goddard Space Flight Center. Prior to this, he was a numerical weather prediction officer in the US Air Force from September 1975 to December 1979. He earned his Ph.D. in meteorology from MIT. Dr. Duffy has written several books on transform methods, engineering mathematics, Green’s functions, and mixed boundary value problems.
Review :
This third edition includes new examples, problems, projects, and, more significantly, new and improved coverage of Green's functions and matrix exponential, numerical methods for solving differential equations, and probability and random processes. ! The text accommodates two general tracks: the differential equations course, and the engineering mathematics course. Duffy's career included 25 years with NASA at the Goddard Space Flight Center (until 2005), and he taught for many years at the US Naval Academy and the US Military Academy. --SciTech Book News, February 2011
