Advanced Engineering Mathematics with MATLAB  book cover
4th Edition

Advanced Engineering Mathematics with MATLAB

ISBN 9781498739641
Published December 22, 2016 by Chapman and Hall/CRC
1004 Pages 104 Color & 239 B/W Illustrations

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Book Description

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


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    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.