Episodes
The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. The program "predprey" studies this model.
Published 05/03/16
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. The program "lorenzgui" studies this model.
Published 05/03/16
Throw a rectangular box with sides of three different lengths into the air. You can get the box to tumble stably about its longest axis or its shortest axis. But if you try to make it tumble about it middle axis, you will find the motion is unstable.
Published 05/03/16
An ODE 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.
Published 05/03/16
The MATLAB documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite.
Published 05/03/16
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.
Published 05/03/16
ODE45 is usually the function of choice among the ODE solvers. It compares 4th and 5th order methods to estimate error and determine step size.
Published 05/03/16
ODE23 compares 2nd and 3rd order methods to automatically choose the step size and maintain accuracy. It is the simplest MATLAB solver that has automatic error estimate and continuous interpolant. ODE23 is suitable for coarse accuracy requirements.
Published 05/03/16
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.
Published 05/03/16
ODE4 implements the classic Runge-Kutta method, 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.
Published 05/03/16
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 trapezoid method.
Published 05/03/16
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.
Published 05/03/16
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.
Published 04/12/16
The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth.
Published 04/12/16
Around every circle, the solution to Laplace’s equation is a Fourier series with coefficients proportional to r^n. On the boundary circle, the given boundary values determine those coefficients.
Published 04/12/16
Even functions use only cosines and odd functions use only sines. The coefficients in the Fourier series come from integrals.
Published 04/12/16
A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions.
Published 04/12/16
Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region.
Published 04/12/16
A second order equation can change from two initial conditions to boundary conditions at two points.
Published 04/12/16
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.
Published 04/12/16
Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.
Published 04/12/16
A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy.
Published 04/12/16
An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues.
Published 04/12/16
If A and B are "similar" then B has the same eigenvalues as A.
Published 04/12/16