Advanced Engineering Mathematics with MATLAB, Third Edition: (Advances in Applied Mathematics)

Taking a practical approach to the subject, Advanced Engineering Mathematics with MATLAB®, Third Edition continues to integrate technology into the conventional topics of engineering mathematics. The author employs MATLAB to reinforce concepts and solve problems that require heavy computation. MATLAB scripts are available for download at www.crcpress.com Along with new examples, problems, and projects, this updated and expanded edition incorporates several significant improvements. New to the Third Edition New chapter on Green’s functions New section that uses the matrix exponential to solve systems of differential equations More numerical methods for solving differential equations, including Adams–Bashforth and finite element methods New chapter on probability that presents basic concepts, such as mean, variance, and probability density functions New chapter on random processes that focuses on noise and other random fluctuations Suitable for a differential equations course or a variety of engineering mathematics courses, the text covers fundamental techniques and concepts as well as Laplace transforms, separation of variable solutions to partial differential equations, the z-transform, the Hilbert transform, vector calculus, and linear algebra. It also highlights many modern applications in engineering to show how these topics are used in practice. A solutions manual is available for qualifying instructors.

Table of Contents:
COMPLEX VARIABLES Complex Numbers Finding Roots The Derivative in the Complex Plane: The Cauchy–Riemann Equations Line Integrals 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 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 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 FOURIER TRANSFORM Fourier Transforms Fourier Transforms Containing the Delta Function Properties of Fourier Transforms Inversion of Fourier Transforms Convolution Solution of Ordinary Differential Equations by Fourier Transforms 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 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 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 The Laplace Transform Method Numerical Solution of the Wave Equation THE HEAT EQUATION Derivation of the Heat Equation Initial and Boundary Conditions Separation of Variables The Laplace Transform Method The Fourier Transform Method The Superposition Integral Numerical Solution of the Heat Equation LAPLACE’S EQUATION Derivation of Laplace’s Equation Boundary Conditions Separation of Variables The Solution of Laplace’s Equation on the Upper Half-Plane Poisson’s Equation on a Rectangle The Laplace Transform Method Numerical Solution of Laplace’s Equation Finite Element Solution of Laplace’s Equation GREEN’S FUNCTIONS What Is a Green’s Function? Ordinary Differential Equations Joint Transform Method Wave Equation Heat Equation Helmholtz’s Equation VECTOR CALCULUS Review Divergence and Curl Line Integrals The Potential Function Surface Integrals Green’s Lemma Stokes’ Theorem Divergence Theorem 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 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 Differential Equations Forced by Random Forcing Two-State Markov Chains Birth and Death Processes Poisson Processes Random Walk ANSWERS TO THE ODD-NUMBERED PROBLEMS INDEX