# Advanced Engineering Mathematics with MATLAB

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## Book Description

Taking a practical approach to the subject,** Advanced Engineering Mathematics with MATLAB ^{®}, 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**

## Author(s)

### Biography

**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.

## Reviews

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