Advanced Engineering Mathematics with MATLAB

Dean G. Duffy is a former mathematics instructor at the US Naval Academy and US Military Academy. He spent 25 years working 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. 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.

CLASSIC ENGINEERING MATHEMATICS

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

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

Vector Calculus
Review
Divergence and Curl
Line Integrals
The Potential Function
Surface Integrals
Green’s Lemma
Stokes’ Theorem
Divergence Theorem

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 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
Numerical Solution of the Wave Equation

The Heat Equation Derivation of the Heat Equation
Initial and Boundary Conditions
Separation of Variables
Numerical Solution of the Heat Equation

Laplace’s Equation Derivation of Laplace’s Equation
Boundary Conditions
Separation of Variables
Poisson’s Equation on a Rectangle
Numerical Solution of Laplace’s Equation
Finite Element Solution of Laplace’s Equation

TRANSFORM METHODS

Complex Variables Complex Numbers
Finding Roots
The Derivative in the Complex Plane: The Cauchy-Riemann Equations
Line Integrals
The 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
Conformal Mapping

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
The Solution of Laplace’s Equation on the Upper Half-Plane
The Solution of the Heat Equation

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 Solution of the Wave Equation
The Solution of the Heat Equation
The Superposition Integral and the Heat Equation
The Solution of Laplace’s Equation

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

Green’s Functions What Is a Green’s Function?
Ordinary Differential Equations
Joint Transform Method
Wave Equation
Heat Equation
Helmholtz’s Equation
Galerkin Methods

STOCHASTIC PROCESSES

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
Two-State Markov Chains
Birth and Death Processes
Poisson Processes

Itˆo’s Stochastic Calculus Random Differential Equations
Random Walk and Brownian Motion
Itˆo’s Stochastic Integral
Itˆo’s Lemma
Stochastic Differential Equations
Numerical Solution of Stochastic Differential Equations