Lær fra MIT, Microsoft, Google og Meta. Helt gratis.

Second Order Systems

Description: An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues. Related section in textbook: 6.5b Instructor: Prof. Gilbert Strang

Positive Definite Matrices

Description: A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy. Related section in textbook: 7.2 Instructor: Prof. Gilbert Strang

Singular Value Decomposition (the SVD)

Description: The SVD factors each matrix into an orthogonal matrix times a diagonal matrix (the singular value) times another orthogonal matrix: Rotation times stretch times rotation. Related section in textbook: 7.2b Instructor: Prof. Gilbert Strang

Boundary Conditions Replace Initial Conditions

Description: A second order equation can change from two initial conditions to boundary conditions at two points. Related section in textbook: 7.3 Instructor: Prof. Gilbert Strang

Laplace Equation

Description: Laplace’s partial differential equation describes temperature distribution inside a circle or a square or any plane region. Related section in textbook: 7.4 Instructor: Prof. Gilbert Strang

Fourier Series

Description: A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions. Related section in textbook: 8.1 Instructor: Prof. Gilbert Strang

Examples of Fourier Series

Description: Even functions use only cosines and odd functions use only sines. The coefficients in the Fourier series come from integrals. Related section in textbook: 8.1b Instructor: Prof. Gilbert Strang

Fourier Series Solution of Laplace's Equation

Description: Around every circle, the solution to Laplace’s equation is a Fourier series with coefficients proportional to rn. On the boundary circle, the given boundary values determine those coefficients. Related section in textbook: 8.1c Instructor: Prof. Gilbert Strang

Heat Equation

Description: The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth. Related section in textbook: 8.3 Instructor: Prof. Gilbert Strang

Wave Equation

Description: The wave equation shows how waves move along the x axis, starting from a given wave shape and its velocity. There can be fixed endpoints as with a violin string. Related section in textbook: 8.4 Instructor: Prof. Gilbert Strang

Euler, ODE1

Descriptions: ODE1 implements Euler’s method. It provides an introduction to numerical methods for ODEs and to the MATLAB® suite of ODE solvers. Exponential growth and compound interest are used as examples. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Midpoint Method, ODE2

Description: ODE2 implements a midpoint method with two function evaluations per step. This method is twice as accurate as Euler’s method. A nonlinear equation defining the sine function provides an example. An exercise involves implementing a related trapezoid method. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Classical Runge-Kutta, ODE4

Description: ODE4 implements the classic Runge-Kutta method, which is the most widely used numerical method for ODEs over the past 100 years. Its major shortcoming is the lack of an error estimate. A simple model of the growth of a flame is an example that is used here and in later videos. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Order, Naming Conventions

Description: The digits in the name of a MATLAB® ODE solver reflect its order and resulting accuracy. A method is said to have order p if cutting the step size in half reduces the error in one step by a factor of two to the power p+1. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Estimating Error, ODE23

Description: ODE23 compares methods of order two and three to automatically choose the step size and maintain a specified accuracy. It is the simplest MATLAB® solver that has modern features such as automatic error estimate and continuous interpolant. ODE23 is suitable for coarse accuracy requirements such as computer graphics. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

ODE45

Description: ODE45 is usually the function of choice among the ODE solvers. It compares methods of orders four and five to estimate error and determine step size. ODE45 is so accurate that its default behavior is to use its interpolant to provide results at intermediate points. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Stiffness, ODE23s, ODE15s

Descriptions: A problem is said to be stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results. The flame model demonstrates stiffness. ODE solvers with names ending in “s”, such as ODE23s and ODE15s, employ implicit methods and are intended for stiff problems. These methods require more work per step, but take many fewer steps. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Systems of Equations

Descriptions: An ordinary differential equation involving higher order derivatives is rewritten as a vector system involving only first order derivatives. The classic Van der Pol nonlinear oscillator is provided as an example. The VdP equation becomes stiff as the parameter is increased. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

The MATLAB ODE Suite

Descriptions: The MATLAB® documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Tumbling Box

Descriptions: Throw a rectangular object with sides of three different lengths (such as a cereal box), into the air. You can get the box to tumble stably about its longest axis, or about its shortest axis. But if you try to make it tumble about it middle axis, you will find the motion is unstable. The model of the angular momentum is a nonlinear system of three differential equations. There are six critical points: the four corresponding to the long and short axes are stable; the two corresponding to the middle axis are unstable. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Predator-Prey Equations

Descriptions: The classic Lotka-Volterra model of predator-prey competition, which describes interactions between foxes and rabbits, or big fish and little fish, is the foundation of mathematical ecology. It has also been applied to many other fields, including economics. The model is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. The one nonzero critical point is stable. All solutions are periodic. The program “predprey” provides an app for studying the model. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

Lorenz Attractor and Chaos

Descriptions: The Lorenz chaotic attractor was discovered by Edward Lorenz in 1963 when he was investigating a simplified model of atmospheric convection. It is a nonlinear system of three differential equations. With the most commonly used values of three parameters, there are two unstable critical points. The solutions remain bounded, but orbit chaotically around these two points. The program “lorenzgui” provides an app for investigating the Lorenz attractor. The resulting 3-D plot looks like a butterfly. Related MATLAB code files can be downloaded from MATLAB Central Instructor: Cleve Moler

MIT D4M: Signal Processing on Databases, Fall 2012

How We Teach: MIT 6.046J Design and Analysis of Algorithms, Spring 2015