Advanced Engineering Mathematics with Mathematica (Hardcover)

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 Exercises 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 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 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 Exercises 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 Exercises 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 Exercises 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 Exercises 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 Exercises 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