Advanced Engineering Mathematics with MATLAB: 4th Edition (e-Book) book cover

Advanced Engineering Mathematics with MATLAB

4th Edition

By Dean G. Duffy

Chapman and Hall/CRC

980 pages | 104 Color Illus. | 239 B/W Illus.

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Advanced Engineering Mathematics with MATLAB, Fourth Edition builds upon three successful previous editions. It is written for today’s STEM (science, technology, engineering, and mathematics) student.

Three assumptions under lie its structure: (1) All students need a firm grasp of the traditional disciplines of ordinary and partial differential equations, vector calculus and linear algebra. (2) The modern student must have a strong foundation in transform methods because they provide the mathematical basis for electrical and communication studies. (3) The biological revolution requires an understanding of stochastic (random) processes. The chapter on Complex Variables, positioned as the first chapter in previous editions, is now moved to Chapter 10.

The author employs MATLAB to reinforce concepts and solve problems that require heavy computation. Along with several updates and changes from the third edition, the text continues to evolve to meet the needs of today’s instructors and students.


  • Complex Variables, formerly Chapter 1, is now Chapter 10.
  • A new Chapter 18: Itô’s Stochastic Calculus.
  • Implements numerical methods using MATLAB, updated and expanded
  • Takes into account the increasing use of probabilistic methods in engineering and the physical sciences
  • Includes many updated examples, exercises, and projects drawn from the scientific and engineering literature
  • Draws on the author’s many years of experience as a practitioner and instructor
  • Gives answers to odd-numbered problems in the back of the book
  • Offers downloadable MATLAB code at
  • Table of Contents


    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


    Cramer’s Rule

    Row Echelon Form and Gaussian Elimination

    Eigenvalues and Eigenvectors

    Systems of Linear Differential Equations

    Matrix Exponential

    Vector Calculus


    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


    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


    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


    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


    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



    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

    About the Author

    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.

    About the Series

    Advances in Applied Mathematics

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    Subject Categories

    BISAC Subject Codes/Headings:
    MATHEMATICS / General
    MATHEMATICS / Applied
    MATHEMATICS / Differential Equations