Advanced Engineering Mathematics with MATLAB

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