Table of Contents

1.1 Introduction

1.2 Polar Form of Complex Numbers

1.3 Limit, Continuity, and Differentiability

1.4 Analytic Functions, Cauchy-Riemann Equations

1.5 Exponential Function

1.6 Trigonometric and Hyperbolic Functions

1.7 Logarithmic Function

Problem Set

2.1 Introduction

2.2 Mathematical Models of Dynamic Systems

2.3 Separable, First-Order ODEs

2.4 Exact, First-Order ODEs

2.5 Linear, First-Order ODEs and Applications

2.6 Orthogonal Trajectories

Problem Set

Chapter 3   Second and Higher-Order, Ordinary Differential Equations           70

3.1 Introduction

3.2 Homogeneous, Second-Order ODEs with Constant Coefficients and Applications

3.3 Second-Order, Euler-Cauchy Differential Equation

3.4 Nonhomogeneous, Linear Second-Order ODEs with Constant Coefficients and Applications

3.5 Method of Variation of Parameters, Green's Function

3.6 Homogeneous, Linear Higher-Order ODEs

3.7 Nonhomogeneous, Linear Higher-Order ODEs

Problem Set

Chapter 4   Series Solution of Ordinary Differential Equations                         106

4.1 Introduction

4.2 Series Solution Method

4.3 Legendre’s Equation and Polynomials

4.4 Frobenius Method

4.5 Bessel’s Equation and Functions

4.6 Differential Equations Satisfied by Bessel Functions

4.7 Orthogonal Functions

Problem Set

Chapter 5   Laplace Transformation                                                                      142

5.1 Introduction

5.2 Special Functions

5.3 Laplace Transform of Derivatives and Integrals

5.4 Inverse Laplace Transformation

5.5 Periodic Functions

5.6 System Response

Table 5.1 Laplace Transform Pairs

Problem Set

Chapter 6   Fourier Analysis                                                                                    180

6.1 Introduction

6.2 Steady-State Response via Fourier Series

6.3 Fourier Integral

6.4 Fourier Transforms

6.5 Inverse Fourier Transformation

Table 6.2  Fourier Transform Pairs

Tables 6.3, 6.4  Fourier Cosine and Sine Transform Pairs

Problem Set

Chapter 7   Partial Differential Equations                                                              220

7.1 Introduction      

7.2 One-Dimensional Wave Equation, Free Vibration of an Elastic String

7.3 Free Vibration of a Uniform Beam

7.4 One-Dimensional Heat Equation, Two-Dimensional Steady-State Heat Flow

7.5 Two-Dimensional Wave Equation, Vibration of Membranes

7.6 Steady-State Heat Conduction in a Solid Sphere

7.7 Solution of PDEs via Laplace Transformation

Problem Set

Chapter 8   Matrix Analysis                                                                                       260

8.1 Introduction    

8.2 Linear Systems of Equations

8.3 Determinant of a Matrix

8.4 Inverse of a Matrix

Problem Set

Chapter 9   Matrix Eigenvalue Problem                                                                   280

9.1 Introduction  

9.2 Eigenvalues of Special Matrices

9.3 Matrix Diagonalization

9.4 Modal Decomposition

9.5 Simultaneous Diagonalization

9.6 Systems of Linear, First-Order Initial-Value Problems

9.7 State Variables

Problem Set

Chapter 10 Introduction to MATLAB^®                                                                    324

10.1 MATLAB Built-in Functions

10.2 Vectors and Matrices

10.3 Symbolic Toolbox

10.4 Program Flow Control

10.5 Displaying Formatted Data

10.6 Plotting

10.7 User-Defined Functions and Script Files

Problem Set

Chapter 11 Numerical Solution of Equations and Systems                                    350

11.1 Introduction to Numerical Methods

Solution of Equations of a Single Variable

11.2 Bracketing Methods

11.3 Fixed-Point Method

11.4 Newton's Method (Newton-Raphson Method)

11.5 Secant Method

11.6 Finding Roots Using the MATLAB Built-in Function fzero

Solution of Linear Systems of Equations

11.7 Gauss Elimination Method

11.8 LU Factorization Methods

11.9 Iterative Solution of Linear Systems

11.10 Ill-Conditioning and Error Analysis

11.11 Systems of Nonlinear Equations

Problem Set

Chapter 12 Curve Fitting and Interpolation                                                             418

12.1 Linear Regression

12.2 Linearization of Nonlinear Data

12.3 Polynomial Regression

12.4 Polynomial Interpolation

12.5 Spline Interpolation

12.6 Fourier Approximation and Interpolation

Problem Set

Chapter 13 Numerical Differentiation and Integration                                            468

13.1 Finite-Difference Formulas for Numerical Differentiation

13.2 Numerical Integration: Newton-Cotes Formulas

13.3 Numerical Integration of Analytical Functions: Gaussian Quadrature

Problem Set

Chapter 14 Numerical Solution of Initial- and Boundary-Value Problems            504

Initial-Value Problems

14.1 Euler's Method

14.2 Runge-Kutta Methods

14.3 Multistep Methods

14.4 Systems of Ordinary Differential Equations

14.5 MATLAB Built-in Functions for Initial-Value Problems

Boundary-Value Problems

14.6 Shooting Method

14.7 Finite-Difference Method

14.8 MATLAB Built-in Function bvp4c for Boundary-Value Problems

Problem Set

Chapter 15 Numerical Methods for the Eigenvalue Problem                                   564    

15.1 Power Method - Estimation of the Dominant Eigenvalue

15.2 Deflation Methods

15.3 Householder Tridiagonalization and QR Factorization Methods

Problem Set

Chapter 16 Numerical Solution of Partial Differential Equations                           592    

16.1 Elliptic Partial Differential Equations

16.2 Parabolic Partial Differential Equations

16.3 Hyperbolic Partial Differential Equations

Problem Set

   Appendix 1      References                                                                         624       

   Appendix 2      Useful Formulas                                                                 625       

   Appendix 3      Tables                                                                                 627       

            Table 1          Gamma Function                                               

            Table 2A       Bessel Functions of the First Kind                   

            Table 2B       Bessel Functions of the Second Kind

           

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