Math 716: Introduction to Partial Differential Equation

• Syllabus for the course.
• Week 1 notes
• Week 2 notes
• 1st Homework Set (Due 1-17-07)
• Week 3 notes
• Homework 1 solutions
• 2nd Homework Set (Due 1-26-07)
• Week 4 notes
• Homework 2 solution
• Week 5 notes
• 3rd Homework Set (Due 2-7-07)
• Week 6 notes
• Homework 3 solution
• 4th Homework Set (Due 2-16-07)
• Week 7 notes
• Homework 4 solution
• Week 8 notes
• 5th Homework Set (Due 2-26-07)
• Homework 5 solution
• Week 9 notes
• 6th Homework Set (Due 3-7-07)
• Week 10 notes
• Homework 6 solution
• Take Home Finals (Strictly follow instruction, Due 3-13-07, 10 a.m.)

Numerical Analysis

Lecture notes

LECTURE 1	LECTURE 5	LECTURE 9
LECTURE 2	LECTURE 6	LECTURE 10
LECTURE 3	LECTURE 7	LECTURE 11
LECTURE 4	LECTURE 8	LECTURE 12

Examples

EXAMPLES 1

EXAMPLES 2

Numerical Analysis - Numerical Methods

Calculus and Fundamentals

1. Calculus Review
2. Big "O" Truncation Error
3. Complex Numbers
4. Complex Functions
5. Using MATLAB for Numerical Analysis

The Solution of Nonlinear Equations f(x) = 0

1. Fixed Point Iteration
2. Bisection Method
3. False Position or Regula Falsi Method
4. Newton-Raphson Method
5. Secant Method
6. Muller's Method
7. Aitken's Method & Steffensen's Acceleration
8. Accelerated & Modified Newton-Raphson
9. Improved Newton Method
10. Halley's Method
11. Horner's Method
12. Lin-Bairstow Method
13. Brent's Method
14. Graeffe's Method
15. Nonlinear Systems
16. Broyden's Method

The Solution of Linear Systems AX = B

1. Triangular Systems and Back Substitution
2. Gauss-Jordan Elimination and Pivoting
3. Tri-Diagonal Matrices
4. Inverse Matrix
5. LU Factorization
6. Cholesky, Doolittle and Crout Factorizations
7. Jacobi and Gauss-Seidel Iteration
8. Successive Over Relaxation - SOR
9. Pivoting Methods
10. Iterative Refinement
11. Row Reduced Echelon Form
12. Homogeneous Linear Systems
13. Kirchoff's Law
14. Leontief Model
15. Linear Programming-Simplex Method

Interpolation and Polynomial Approximation

1. Maclaurin and Taylor Series
2. Lagrange Polynomial Interpolation and Approximation
3. Newton Interpolation Polynomial
4. Hermite Polynomial Interpolation
5. Cubic Splines
6. B-Splines
7. Bézier Curves Bézier Curves
8. Chebyshev Approximation Polynomial
9. Pade Approximation
10. Rational Approximation
11. Aitken's and Neville's Interpolation
12. Legendre Polynomials
13. The Tangent Parabola
14. Catenary

Curve Fitting

1. Least Squares Lines
2. Least Squares Polynomials
3. Nonlinear Curve Fitting
4. Logistic Curve
5. FFT and Trigonometric Polynomials
6. Conic Fit
7. Circle of Curvature

Numerical Differentiation

1. Numerical Differentiation
2. Richardson Extrapolation
3. Derive Numerical Differentiation Formulae

Numerical Integration

1. Riemann Sums
2. Midpoint Rule
3. Newton-Cotes Integration
4. Trapezoidal Rule for Numerical Integration
5. Simpson's Rule for Numerical Integration
6. Simpson's 3/8 Rule for Numerical Integration
7. Boole's Rule
8. Romberg Integration
9. Adaptive Simpson's Rule
10. Gauss-Legendre Quadrature
11. Cubic Spline Quadrature
12. Monte Carlo Pi
13. Monte Carlo Integration
14. 2D Trapezoidal and Simpson Rules

Solution of Differential Equations

1. Euler's Method for ODE's
2. Taylor Series Method for ODE's
3. Runge-Kutta Method
4. Runge-Kutta-Fehlberg Method
5. Adams-Bashforth-Moulton Method
6. Milne-Simpson's Method
7. Predictor-Corrector Methods
8. Shooting Methods for ODE's
9. Finite Difference Method for ODE's
10. Galerkin's Method
11. Painleve Property
12. Lotka-Volterra Model
13. Pendulum
14. Projectile Motion
15. Lorenz Attractor
16. van der Pol System
17. Harvesting Model
18. Frobenius Series Solution
19. Picard Iteration
20. Spring-Mass Systems

Solution of Partial Differential Equations

1. Finite Difference Method
2. Crank-Nicolson Method
3. Elliptic PDE's

Eigenvalues and Eigenvectors

1. Eigenvalues and Eigenvectors
2. Power method
3. Jacobi method
4. Householder Transformations
5. QR method
6. Compartment Model
7. Earthquake Model
8. Matrix Exponential
9. Faddeev-Leverrier Method
10. Hessenberg Factorization

Numerical Optimization

1. Golden Ratio Search
2. Fibonacci Search
3. Quadratic Interpolative Search
4. Nelder Mead Method
5. Powell's Method
6. Steepest Descent - Gradient Search
7. Newton's Search Method

SOS Mathematics

First Order Differential Equations

• Linear Equations
• Separable Equations
• Qualitative Technique: Slope Fields
• Equilibria and the Phase Line
• Bifurcations
• Bernoulli Equations
• Riccati Equations
• Homogeneous Equations
• Exact and Non-Exact Equations
• Integrating Factor technique
• Some Applications
  • Radioactive Decay
  • Newton's Law of Cooling
  • Orthogonal Trajectories
  • Population Dynamics
• Numerical Technique: Euler's Method
• Existence and Uniqueness of Solutions
• Picard Iterative Process

Second Order Differential Equations

• Nonlinear Equations
• Linear Equations
• Homogeneous Linear Equations
• Linear Independence and the Wronskian
• Reduction of Order
• Homogeneous Equations with Constant Coefficients
• Non-Homogeneous Linear Equations
  • Method of Undetermined Coefficients
  • Method of Variation of Parameters
• Euler-Cauchy Equations
• Series Solutions
  • Introduction
  • Derivatives and Index Shifting
  • First Examples
  • Airy's Equation
  • The Radius of Convergence of Series Solutions
  • Hermite's Equation

Higher Order Linear Equations

• Introduction and Basic Results
• Homogeneous Linear Equations with Constant Coefficients
• Non-Homogeneous Linear Equations
• Method of Undetermined Coefficients
• Method of Variation of Parameters

Laplace Transform

• Basic Definitions and Results
• Application to Differential Equations
• Impulse Functions: Dirac Function
• Convolution Product
• Table of Laplace Transforms

Systems of Differential Equations

• Introduction and Motivation
• Second Order Equations and Systems
• Euler's Method for Systems
• Qualitative Analysis
• Linear Systems
  • Introduction and First Definitions
  • Vector Representations of Solutions of Linear Systems
  • Eigenvalues and Eigenvectors Technique
    • Real Eigenvalues
    • Repeated Eigenvalues
    • Complex Eigenvalues
    • Zero Eigenvalues
  • Qualitative Analysis of Linear Systems
• Nonlinear Systems
  • Equilibrium Point Analysis: Linearization Technique

Fourier Series

• Fourier Series: Basic Results
• Fourier Sine and Cosine Series
• Convergence of Fourier Series
• More on Convergence of Fourier Series
• Gibbs Phenomenon
• Bessel's Inequality and Parseval Formula: The Energy Theorem
• Operations on Fourier Series
• Application of Fourier Series to Differential Equations

Introduction.

This blog is for notes, textbooks and other references for two courses that I am taking.