Numerical Methods for Engineers 

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Course Overview

Week One Introduction

Binary Numbers

Lecture 1

Double Precision

Lecture 2

MATLAB as a Calculator

Lecture 3

Scripts and Functions

Lecture 4

Vectors

Lecture 5

Line Plots

Lecture 6

Matrices

Lecture 7

Logicals

Lecture 8

Conditionals

Lecture 9

Loops

Lecture 10

Logistic Map (Part A)

Lecture 11

Logistic Map (Part B)

Lecture 12

Welcome and Course Information

How to Write Math in the Discussions Using MathJax

MATLAB Online

Rounding Binary Numbers

Computer numbers

REALMAX

REALMIN

EPS

Logical Expressions

Logical Vectors

Quadratic Equation

Background for the Logistic Map

Period-2

Diagnostic Quiz

Week One Assessment

Week Two Introduction

Bisection Method

Lecture 13

Newton's Method

Lecture 14

Secant Method

Lecture 15

Order of Convergence

Lecture 16

Convergence of Newton's Method

Lecture 17

Fractals from Newton's Method

Lecture 18

Coding the Newton Fractal

Lecture 19

Root-Finding in MATLAB

Lecture 20

Feigenbaum Delta (Part A)

Lecture 21

Feigenbaum Delta (Part B)

Lecture 22

Feigenbaum Delta (Part C)

Lecture 23

Estimate the Square-root of Three Using the Bisection Method

Estimate the Square-root of Three Using Newton's Method

Estimate the Square-Root of Three Using the Secant Method

Rates of Convergence

Order of Convergence of the Secant Method

The Four Fourth Roots of Unity

Compute the Value of m in the Period-Two Cycle

Week Two Assessment

Week Three Introduction

Gaussian Elimination without Pivoting

Lecture 24

Gaussian Elimination with Partial Pivoting

Lecture 25

LU Decomposition with Partial Pivoting

Lecture 26

Operation Counts

Lecture 27

Operation Counts for Gaussian Elimination

Lecture 28

Operation Counts for Forward and Backward Substitution

Lecture 29

Eigenvalue Power Method

Lecture 30

Eigenvalue Power Method (Example)

Lecture 31

Matrix Algebra in MATLAB

Lecture 32

Systems of Nonlinear Equations

Lecture 33

Systems of Nonlinear Equations (Example)

Lecture 34

Fractals from the Lorenz Equations

Lecture 35

Round-off Errors in Gaussian Elimination

Reduced Round-off Errors in Gaussian Elimination with Partial Pivoting

The (PL)U Decomposition of A

Estimating Computational Time using Operation Counts

Summation Identities

Operation Counts for a Lower Triangular System

Convergence of the Eigenvalue Power Method

Determine the Dominant Eigenvalue

How to Solve Three Nonlinear equations

Week Three Assessment

Week Four Introduction

Midpoint Rule

Lecture 36

Trapezoidal Rule

Lecture 37

Simpson's Rule

Lecture 38

Composite Quadrature Rules

Lecture 39

Gaussian Quadrature

Lecture 40

Adaptive Quadrature

Lecture 41

Quadrature in MATLAB

Lecture 42

Interpolation

Lecture 43

Cubic Spline Interpolation (Part A)

Lecture 44

Cubic Spline Interpolation (Part B)

Lecture 45

Interpolation in MATLAB

Lecture 46

Bessel Functions and their Zeros

Lecture 47

The Midpoint Rule is the Area of a Rectangle

Midpoint Rule for a Quadratic Function

Derive the Trapezoidal Rule

Derive Simpson's Rule

Simpson's 3/8 Rule

Three-point Legendre-Gauss Quadrature

Computing the Error in an Adaptive Quadrature

Linear and Quadratic Interpolation

Cubic Spline Interpolation with Endpoint Slopes Known

Cubic Spline Interpolation with the Not-a-Knot Condition

Week Four Assessment

Week Five Introduction

Euler Method

Lecture 48

Modified Euler Method

Lecture 49

Runge-Kutta Methods

Lecture 50

Second-Order Runge-Kutta Methods

Lecture 51

Higher-Order Runge-Kutta Methods

Lecture 52

Higher-Order ODEs and Systems

Lecture 53

Adaptive Runge-Kutta Method

Lecture 54

Integrating ODEs in MATLAB (Part A)

Lecture 55

Integrating ODEs in MATLAB (Part B)

Lecture 56

Shooting Method for Boundary Value Problems

Lecture 57

The Two-Body Problem (Part A)

Lecture 58

The Two-Body Problem (Part B)

Lecture 59

When the Euler Method is Exact

When the Modified Euler Method is Exact

Ralston's Method

Runge-Kutta Methods and Quadrature Formulas

Fourth-Order Runge-Kutta Method and Simpson's Rule

Systems of ODEs

Example of Adaptive Integration

Circular orbits

Week Five Assessment

Week Six Introduction

Boundary and Initial Value Problems

Lecture 60

Central Difference Approximation

Lecture 61

Discrete Laplace Equation

Lecture 62

Natural Ordering

Lecture 63

Matrix Formulation

Lecture 64

MATLAB Solution of the Laplace Equation (Direct Method)

Lecture 65

Jacobi, Gauss-Seidel and SOR Methods

Lecture 66

Red-Black Ordering

Lecture 67

MATLAB Solution of the Laplace Equation (Iterative Method)

Lecture 68

Explicit Methods for Solving the Diffusion Equation

Lecture 69

Von Neumann Stability Analysis of the FTCS Scheme

Lecture 70

Implicit Methods for Solving the Diffusion Equation

Lecture 71

Crank-Nicolson Method for the Diffusion Equation

Lecture 72

MATLAB Solution of the Diffusion Equation

Lecture 73

Two-Dimensional Diffusion Equation

Lecture 74

Concluding Remarks

Higher-order Central Difference Approximation

Mean Value Property of the Laplace Equation

Coordinates of the four corners

The Discrete Laplace Equation on a Four-by-Four Grid

Number of Interior and Boundary Points

Iterative Solution of a System of Linear Equations

Using a Second-Order Time-Stepping Method

FTCS Scheme for the Advection Equation

Von Neumann Stability Analysis of the FTCS Scheme for the Advection Equation

Implicit Discrete Advection Equation

Lax Scheme for the Advection Equation

Difference Approximations for the Derivative at Boundary Points

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Acknowledgements

Classify Partial Differential Equations

Week Six Assessment