Advanced Engineering Mathematics with Mathematica: 1st Edition (Hardback) book cover

Advanced Engineering Mathematics with Mathematica

1st Edition

By Edward B. Magrab

CRC Press

632 pages | 85 B/W Illus.

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Hardback: 9780367893255
pub: 2020-03-16
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Advanced Engineering Mathematics with Mathematica presents advanced analytical solution methods that are used to solve boundary value problems in engineering and integrates these methods with over 200 Mathematica programs. It emphasizes the Sturm-Liouville system and the generation and application of orthogonal functions, which are used by the separation of variables method to solve partial differential equations. It introduces the relevant aspects of complex variables, matrices and determinants, Fourier series and transforms, solution techniques for ordinary differential equations, the Laplace transform, and procedures to make ordinary and partial differential equations used in engineering non-dimensional. To show the diverse application of the material, a large number of solved boundary value problems are presented.

Table of Contents


1 Matrices, Determinants, And Systems of Equations

1.1 Definitions

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


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


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


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 Coefficients

4.2.3 Reduction of Order

4.2.4 Cauchy-Euler Equation

4.2.5 Particular Solutions: Method of Undetermined Coefficients

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


5 Ordinary Differential Equations Part II – Power Series Solutions

5.1 Power Series Solutions to Ordinary Differential Equations

5.1.1 Classification of Singularities

5.1.2 Power Series Solution About an Ordinary Point

5.1.3 Power Series Solution 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 Modified 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


Appendix 5.1 Bessel Function of the Second Kind

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 Example of Sturm-Liouville Equations

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


7 Partial Differential Equations

7.1 Introduction to Second-Order Partial Differential Equations

7.1.2 Classification of Linear Second Order Partial Differential Equations

7.1.1 Representative Application Areas

7.2 Separation of Variables and the Solutions 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 The Diffusion Equation

7.2.5 The Wave Equation

7.2.6 The Poisson Equation

7.2.7 The Bi-harmonic Equation

7.3 Placing Partial Differential Equations into Non-Dimensional Form

7.4 Partial Differential Equation with Irregular Geometries: Numerical Solutions Using Mathematica’s Finite Element Capability

Mathematica Procedures


8 Laplace Transforms

8.1 Laplace Transform

8.1.1 Definition

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

Appendix 8.1 Laplace Transform Pairs

Mathematica Procedures


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 Based-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

B.1 Delta Function

B.2 Delta Function and Heaviside Function

B.3 Delta Function in Two and Three Dimensions and in Different Coordinate Systems

Appendix C Gamma Function

About the Author

Edward B. Magrab is Emeritus Professor in the Department of Mechanical Engineering at the University of Maryland at College Park. He has extensive experience in analytical and experimental analysis of vibrations and acoustics, served as an engineering consultant to numerous companies, and authored or co-authored a number of books on vibrations, noise control, instrumentation, integrated product design, MATLAB®, and Mathematica®. He is a Life Fellow of the American Society of Mechanical Engineers.

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