Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler

Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler

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Gilbert Strang and Cleve Moler provide an overview to their in-depth video series about differential equations and the MATLAB ODE suite.

Gilbert Strang, Cleve Moler


    • May 3, 2016 LATEST EPISODE
    • infrequent NEW EPISODES
    • 14m AVG DURATION
    • 68 EPISODES


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    Latest episodes from Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler

    Predator-Prey Equations

    Play Episode Listen Later May 3, 2016 14:17


    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.

    Lorenz Attractor and Chaos

    Play Episode Listen Later May 3, 2016 10:24


    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.

    Tumbling Box

    Play Episode Listen Later May 3, 2016 9:51


    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.

    Systems of Equations

    Play Episode Listen Later May 3, 2016 14:16


    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.

    The MATLAB ODE Suite

    Play Episode Listen Later May 3, 2016 5:34


    The MATLAB documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite.

    ODE45

    Play Episode Listen Later May 3, 2016 6:46


    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.

    Stiffness, ODE23s, ODE15s

    Play Episode Listen Later May 3, 2016 7:14


    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.

    Order, Naming Conventions

    Play Episode Listen Later May 3, 2016 5:26


    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.

    Estimating Error, ODE23

    Play Episode Listen Later May 3, 2016 10:37


    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.

    Midpoint Method, ODE2

    Play Episode Listen Later May 3, 2016 6:46


    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.

    Classical Runge-Kutta, ODE4

    Play Episode Listen Later May 3, 2016 9:38


    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.

    Euler, ODE1

    Play Episode Listen Later May 3, 2016 15:22


    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.

    Laplace Equation

    Play Episode Listen Later Apr 12, 2016 13:17


    Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region.

    Fourier Series

    Play Episode Listen Later Apr 12, 2016 16:35


    A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions.

    Examples of Fourier Series

    Play Episode Listen Later Apr 12, 2016 13:56


    Even functions use only cosines and odd functions use only sines. The coefficients in the Fourier series come from integrals.

    Fourier Series Solution of Laplace's Equation

    Play Episode Listen Later Apr 12, 2016 14:03


    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.

    Heat Equation

    Play Episode Listen Later Apr 12, 2016 10:48


    The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth.

    Wave Equation

    Play Episode Listen Later Apr 12, 2016 15:13


    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.

    Boundary Conditions Replace Initial Conditions

    Play Episode Listen Later Apr 12, 2016 17:03


    A second order equation can change from two initial conditions to boundary conditions at two points.

    Singular Value Decomposition (the SVD)

    Play Episode Listen Later Apr 12, 2016 14:10


    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.

    Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors

    Play Episode Listen Later Apr 12, 2016 15:54


    Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.

    Second Order Systems

    Play Episode Listen Later Apr 12, 2016 16:50


    An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues.

    Positive Definite Matrices

    Play Episode Listen Later Apr 12, 2016 21:40


    A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy.

    The Matrix Exponential

    Play Episode Listen Later Apr 12, 2016 15:32


    The shortest form of the solution uses the matrix exponential multiplying the starting vector (the initial condition).

    Similar Matrices

    Play Episode Listen Later Apr 12, 2016 14:50


    If A and B are "similar" then B has the same eigenvalues as A.

    Eigenvalues and Eigenvectors

    Play Episode Listen Later Apr 12, 2016 19:01


    The eigenvectors remain in the same direction when multiplied by the matrix. Subtracting an eigenvalue from the diagonal leaves a singular matrix: determinant zero. An n by n matrix has n eigenvalues.

    Diagonalizing a Matrix

    Play Episode Listen Later Apr 12, 2016 11:36


    A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λ is the eigenvalue matrix.

    Powers of Matrices and Markov Matrices

    Play Episode Listen Later Apr 12, 2016 17:53


    Diagonalizing a matrix also diagonalizes all its powers.

    Solving Linear Systems

    Play Episode Listen Later Apr 12, 2016 15:47


    An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combinations of those solutions lead to all solutions.

    Incidence Matrices of Graphs

    Play Episode Listen Later Apr 12, 2016 19:50


    The incidence matrix has a row for every edge, containing -1 and +1 to show which two nodes are connected by that edge.

    The Column Space of a Matrix

    Play Episode Listen Later Apr 12, 2016 12:43


    Capturing all combinations of the columns gives the column space of the matrix. It is a subspace (such as a plane).

    Independence, Basis, and Dimension

    Play Episode Listen Later Apr 12, 2016 13:19


    Vectors are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension = number of basis vectors.

    The Big Picture of Linear Algebra

    Play Episode Listen Later Apr 12, 2016 15:57


    A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.

    Graphs

    Play Episode Listen Later Apr 12, 2016 15:26


    A graph has nodes connected by edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.

    Linearization at Critical Points

    Play Episode Listen Later Apr 12, 2016 15:08


    A critical point is a constant solution to the differential equation. The slope of the right hand side decides stability or instability.

    Linearization of Two Nonlinear Equations

    Play Episode Listen Later Apr 12, 2016 21:40


    With two equations, the two linearized equations use the 2 by 2 matrix of partial derivatives of the right hand sides.

    Eigenvalues and Stability: 2 by 2 Matrix, A

    Play Episode Listen Later Apr 12, 2016 19:30


    Two equations with a constant matrix are stable (solutions approach zero) when the trace is negative and the determinant is positive.

    The Tumbling Box in 3-D

    Play Episode Listen Later Apr 12, 2016 22:53


    A box in the air can rotate stably around its shortest and longest axes. Around the middle axis it tumbles wildly.

    Pictures of Solutions

    Play Episode Listen Later Apr 12, 2016 21:00


    The direction field has an arrow with slope at each point coming from the differential equation. Arrows with the same slope lie along an "isocline".

    Phase Plane Pictures: Source, Sink, Saddle

    Play Episode Listen Later Apr 12, 2016 18:25


    Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exponent or eigenvalue.

    Unforced Damped Motion

    Play Episode Listen Later Apr 12, 2016 14:03


    With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation.

    Impulse Response and Step Response

    Play Episode Listen Later Apr 12, 2016 16:01


    The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.

    Phase Plane Pictures: Spirals and Centers

    Play Episode Listen Later Apr 12, 2016 13:46


    Imaginary exponents with pure oscillation provide a "center" in the phase plane. The point (position, velocity) travels forever around an ellipse.

    Two First Order Equations: Stability

    Play Episode Listen Later Apr 12, 2016 10:31


    A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular).

    Variation of Parameters

    Play Episode Listen Later Apr 12, 2016 19:22


    Combine null solutions to find a particular solution for any right hand side. But it may involve a difficult integral.

    Laplace Transform: First Order Equation

    Play Episode Listen Later Apr 12, 2016 22:37


    Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution.

    Laplace Transforms and Convolution

    Play Episode Listen Later Apr 12, 2016 10:28


    When the input force is an impulse, the output is the impulse response. For all inputs the response is a "convolution" with the impulse response.

    Electrical Networks: Voltages and Currents

    Play Episode Listen Later Apr 12, 2016 16:33


    Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance).

    Method of Undetermined Coefficients

    Play Episode Listen Later Apr 12, 2016 16:31


    With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.

    An Example of Undetermined Coefficients

    Play Episode Listen Later Apr 12, 2016 15:49


    This method is also successful for forces and solutions equal to polynomials times exponentials. Substitute into the equation!

    Laplace Transform: Second Order Equation

    Play Episode Listen Later Apr 12, 2016 16:30


    The algebra problem involves the transfer function. The poles of that function are all-important.

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