### Summary

*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. ** **

**Features:**

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 www.crcpress.com
### Table of Contents

**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