Table of Contents

Chapter 1 Ordinary Differential Equations – Symbolic Solutions

• Verifying Solutions of First-Order ODE's
• Solving ODE's and IVP's and Plotting the Solution
• Orthogonal Trajectories
• Second-Order ODE's
• Series Solutions of ODE's
• Legendre's Equation and Polynomials
• Bessel's Equation and Functions

Chapter 2 Laplace Transformation

• Laplace Transform
• Transfer Function
• Poles and Zeros
• Impulse and Step Responses
• Functions Defined Piecewise
• Response using Transfer Function
• Initial and Final-Value Theorems
• Periodic Functions

Chapter 3 Fourier Analysis

• Fourier Series
• Symbolic Fourier Coefficients
• Extension Beyond One Period
• Even and Odd Functions
• Even and Odd Periodic Extensions
• Fourier Transformation
• Using Fourier Transformation to Solve ODE's

Chapter 4 Partial Differential Equations

• One-Dimensional Wave Equation; Free Vibration of an Elastic String Animation
• Transverse Vibration of a Beam
• One-Dimensional Heat Equation
• Two-Dimensional Heat Equation
• Two-Dimensional Wave Equation; Vibration of Rectangular Membranes
• Free Lateral Vibration of Circular Membranes

Chapter 5 Complex Numbers, Variables, and Functions

• Complex Symbolic Variables
• Roots of a Complex Number
• Complex Equations
• Limit, Continuity and Differentiability
• Analytic Functions
• Regions in the Complex Plane
• Mapping by Elementary Functions

Chapter 6 Complex Integrals and Series

• Complex Integrals
• Parametric Representation of the Path
• Integral Evaluation by Parametric Representation
• Complex Series
• Tests of Convergence
• General Form of Complex Series

Chapter 7 Residue Theory

• Residues
• Integral Evaluation via Residue Theory
• Improper Real Integrals

Chapter 8 Conformal Mapping

• Points, Lines and Curves in the Complex Plane
• Plotting Points, Lines, Circles, Circular Arcs and Polygons
• Measuring Interior Angles
• Plotting Axes and Planes
• Conformal Mapping
• The Bilinear Transformation
• Critical and Fixed Points of Mappings
• Finding Mapping Based on Given Fixed Points
• Finding Mapping from Images
• Mapping Planes to Circles

Chapter 9 Linear Algebra – Vectors and Matrices

• Vectors
• Accessing and Assigning Vector Elements
• Matrices
• Multiplying Matrices
• Linear Indexing
• Multidimensional Matrices
• Matrix Addition and Multiplication
• Matrix Inverse and Matrix Divisions
• Exponents of Matrices
• Dot and Cross Products
• Determinant, Condition Number
• Rank

Chapter 10 Matrix Eigenvalue Problem

• Similarity Transformations
• Generalized Eigenvectors
• Modal Decomposition
• Systems of Linear, First-Order Differential Equations
• Simultaneous Diagonalization
• Application: Undamped Dynamic Systems

Chapter 11 Fundamentals of Numerical Methods

• Bisection Method
• Newton's Method
• Roots of Polynomials
• Roots of Functions
• Polynomial Approximation
• One-, Two-, and Higher-Dimensional Interpolation
• Scattered Data
• Fourier Approximation and Interpolation
• Fast Fourier Transform (FFT)
• Numerical Differentiation
• Numerical Integration

Chapter 12 Numerical Methods for Ordinary Differential Equations

• Setting ODE Solver Options
• Numerical Methods for Solving ODE's
• Fourth-Order Runge-Kutta Method
• Multi-step Methods - Predictor-Corrector Methods
• Systems of First-Order ODE's

Chapter 13 Numerical Methods for Partial Differential Equations

• Elliptic Equations
• Dirichlet Problem; Difference Equations
• Peaceman and Rachford Alternating Direction Implicit (PRADI) Method
• Parabolic Equations
• Finite Difference Method
• Crank-Nicolson Method
• Hyperbolic Equations

Chapter 14 Numerical Methods in Linear Algebra

• Direct Methods for Solving Linear Systems of Equations
• Gauss Elimination with Partial Pivoting
• Vector and Matrix Norms
• LU-Factorization
• Doolittle's Method, Cholesky's Method
• Iterative Methods for Solving Linear Systems of Equations
• Jacobi and Gauss-Seidel Iterations
• Curve Fit; Method of Least Squares
• Overdetermined Linear Systems
• Localization of Eigenvalues
• Matrix Eigenvalue Approximation; Power Method
• Wielandt's Deflation Methods
• Householder and Lanczos Tridiagonalization Methods + QR Factorization

Chapter 15 Probability and Statistics

• Set Operations
• Permutations, Combinations
• Discrete and continuous Random Variables and Probability Distributions
• Random Number Generators
• Normal Distribution
• Joint Probability Distributions
• Statistical Analysis
• Binomial Distribution
• Hypergeometric Distribution
• Poisson Distribution
• Expected Values
• Moment-Generating Function

Index