Advanced Engineering Mathematics with Mathematica

✍ Scribed by Edward B. Magrab

• Publisher: CRC Press
• Year: 2020
• Tongue: English
• Leaves: 551
• Edition: 1
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Chapter 1 Matrices, Determinants, and Systems of Equations
1.1 Defnitions
1.2 Matrix Operations
1.3 Determinants
1.4 Matrix Inverse
1.5 Properties of Matrix Products
1.6 Eigenvalues of a Square Matrix
1.7 Solutions to a System of Equations: Eigenvalues, Eigenvectors,
and Orthogonality
Mathematica Procedures
Exercises
Chapter 2 Introduction to Complex Variables
2.1 Complex Numbers
2.2 Complex Exponential Function: Euler’s Formula
2.3 Analytic Functions
2.3.1 Cauchy–Riemann Conditions
2.3.2 Cauchy Integral Formula
Mathematica Procedures
Exercises
Chapter 3 Fourier Series and Fourier Transforms
3.1 Fourier Series
3.2 Fourier Series in the Frequency Domain
3.3 Fourier Transform
3.3.1 An Intuitive Approach
3.3.2 Fourier Transform
3.3.3 Properties of the Fourier Transform
3.3.4 Convolution Integral
3.3.5 Delta Function
3.4 Fourier Transform and Signal Analysis
3.4.1 Sampling
3.4.2 Aliasing
3.4.3 Short-Time Fourier Transform (STFT
3.4.4 Windowing: The Hamming Window
Mathematica Procedures
Exercises
Chapter 4 Ordinary Differential Equations Part I: Review of First- and
Second-Order Equations
4.1 First-Order Ordinary Differential Equations
4.1.1 Special Cases of First-Order Ordinary Differential
Equations
4.1.2 Bernoulli Equation
4.1.3 Direction Fields
4.2 Second- and Higher-Order Ordinary Differential Equations
4.2.1 Introduction
4.2.2 Homogeneous Differential Equations with Constant
Coeffcients
4.2.3 Reduction of Order
4.2.4 Cauchy–Euler Equation
4.2.5 Particular Solutions: Method of Undetermined
Coeffcients
4.2.6 Particular Solutions: Variation of Parameters
4.2.7 Conversion to a System of First-Order Differential
Equations
4.2.8 Orthogonal Functions and the Solutions to a System of
Second-Order Equations
4.2.9 Making Differential Equations Non-Dimensional
4.2.10 Nonlinear Differential Equations: A Few Special Cases
4.2.11 Phase Plane and Direction Fields
Mathematica Procedures
Exercises
Chapter 5 Ordinary Differential Equations Part II: Power Series Solutions
5.1 Power Series Solutions to Ordinary Differential Equations
5.1.1 Classifcation of Singularities
5.1.2 Power Series Solution about an Ordinary Point
5.1.3 Power Series about a Regular Singular Point: Method
of Frobenius
5.1.4 Bessel’s Equation and Bessel Functions
5.1.5 Derivatives and Integrals of Bessel Functions of the
First and Second Kind
5.1.6 Spherical Bessel Functions
5.1.7 Modifed Bessel Functions
5.1.8 Differential Equations Whose Solutions Are in Terms of
Bessel Functions
5.1.9 Legendre’s Equation and Legendre Polynomials
5.1.10 Associated Legendre’s Equation and Legendre
Polynomials
5.1.11 Hypergeometric Equation and Hypergeometric
Functions
Mathematica Procedures
Exercises
Appendix 5.1
Bessel Function of the Second Kind
Chapter 6 Ordinary Differential Equations Part III: Sturm–Liouville Equation
6.1 Sturm–Liouville Equation
6.1.1 Preliminaries: Adjoint Equations
6.1.2 Sturm–Liouville Equation
6.1.3 Examples of Sturm–Liouville Systems
6.1.4 Orthogonal Functions: Their Generation and
Their Properties
6.1.5 Fourth-Order Sturm–Liouville Differential Equation
6.1.6 General Solution to Nonhomogeneous Sturm–Liouville
Equations
6.2 Orthogonal Functions for Coupled Systems: Two
Dependent Variables
Exercises
Chapter 7 Partial Differential Equations
7.1 Introduction to Second-Order Partial Differential Equations
7.1.1 Classifcation of Linear Second-Order Partial
Differential Equations
7.1.2 Representative Application Areas
7.2 Separation of Variables and the Solution to Partial Differential
Equations of Engineering and Physics
7.2.1 Introduction
7.2.2 Laplace Equation
7.2.3 Helmholtz Equation
7.2.4 Diffusion Equation
7.2.5 Wave Equation
7.2.6 Poisson Equation
7.2.7 Bi-Harmonic Equation
7.3 Placing Partial Differential Equations into Non-Dimensional
Form
7.4 Partial Differential Equations and Irregularly Shaped
Regions: Numerical Solutions Using Mathematica’s Finite
Element Capability
Mathematica Procedures
Exercises
Chapter 8 Laplace Transforms
8.1 Laplace Transform
8.1.1 Defnition
8.1.2 Derivation of Laplace Transform Pairs
8.1.3 Partial Fractions
8.1.4 Convolution Integral
8.1.5 Translation and Scaling
8.1.6 Periodic Functions
8.1.7 Inversion Integral Revisited
8.2 Applications of the Laplace Transform to Ordinary and Partial
Differential Equations
Mathematica Procedures
Exercises
Appendix 8.1
Laplace Transform Pairs
Chapter 9 Putting It All Together—Examples from the Literature
9.1 Introduction
9.2 Squeeze Film Air Damping
9.2.1 Introduction
9.2.2 Squeeze Film Damping for Parallel Rectangular Surfaces
Subject to Harmonic Excitation
9.2.3 Base-Excited Single-Degree-of-Freedom System with
Squeeze Film Air Damping
9.3 Viscous Fluid Damping
9.3.1 Forces on a Submerged Harmonically Oscillating Rigid
Cylinder in a Viscous Fluid
9.3.2 Mass-Excited Single-Degree-of-Freedom System Subject
to Viscous Fluid Damping
9.4 Natural Frequencies of a Cantilever Beam with an In-Span
Spring–Mass System
9.4.1 Introduction
9.4.2 Determination of Natural Frequencies and
Mode Shapes
9.5 Piezoelectric Energy Harvester: Single-Degree-of-Freedom
System
9.5.1 Piezoelectric Generator
9.5.2 Maximum Average Power of a Piezoelectric Generator
9.6 Determination of the Onset of Flutter
9.6.1 Governing Equations
9.6.2 Determination of Flutter Frequencies
9.7 Thermal Runaway in Microwave Heating of Ceramics
9.7.1 Introduction
9.7.2 Heat Equation and Boundary Conditions
9.7.3 Steady-State Microwave Heating of a Slab
9.7.4 Outline to Obtain Numerical Results
Appendix A Series Expansions
Appendix B Delta Function
Appendix C Gamma Function
Bibliography
Index