Learn Differential Equations: Tutorials with Gilbert Strang and Cleve Moler
Video Lectures
Displaying all 68 video lectures.
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Introduction to Differential Equations and the MATLAB® ODE Suite Gilbert Strang and Cleve Moler provide an overview to their in-depth video series about differential equations and the MATLAB® ODE suite. |
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Overview of Differential Equations Differential equations connect the slope of a graph to its height. Slope = height, slope = -height, slope = 2t times height: all linear. Slope = (height)^2 is nonlinear. |
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The Calculus You Need The sum rule, product rule, and chain rule produce new derivatives from known derivatives. The Fundamental Theorem of Calculus says that the integral inverts the derivative. |
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Response to Exponential Input For a linear equation with exponential input from outside and exponential growth from inside, the solution is a combination of two exponentials. |
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Response to Oscillating Input A linear equation with oscillating input has an oscillating output with the same frequency (and a phase shift). |
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Solution for Any Input To solve a linear first order equation, multiply each input by its growth factor and integrate those outputs. |
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Step Function and Delta Function A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump. |
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Response to Complex Exponential For linear equations, the solution for a cosine input is the real part of the solution for a complex exponential input. That complex solution has magnitude G (the gain). |
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Integrating Factor for Constant Rate The integrating factor multiplies the differential equation to allow integration. |
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Integrating Factor for a Varying Rate The integral of a varying interest rate provides the exponent in the growing solution (the bank balance). |
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The Logistic Equation When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state. |
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The Stability and Instability of Steady States Steady state solutions can be stable or unstable – a simple test decides. |
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Separable Equations Separable equations can be solved by two separate integrations, one in t and the other in y. |
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Second Order Equations For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency. |
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Forced Harmonic Motion When the forcing is a sinusoidal input, like a cosine, one particular solution has the same form. But if the forcing frequency equals the natural frequency there is resonance. |
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Unforced Damped Motion With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation. |
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Impulse Response and Step Response 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. |
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Exponential Response — Possible Resonance Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside. |
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Second Order Equations with Damping A damped forced equation has a sinusoidal solution with exponential decay. The damping ratio provides insight into the null solutions. |
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Electrical Networks: Voltages and Currents Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance). |
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Method of Undetermined Coefficients With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form. |
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An Example of Undetermined Coefficients This method is also successful for forces and solutions equal to polynomials times exponentials. Substitute into the equation! |
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Variation of Parameters Combine null solutions to find a particular solution for any right hand side. But it may involve a difficult integral. |
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Laplace Transform: First Order Equation Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution. |
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Laplace Transform: Second Order Equation The algebra problem involves the transfer function. The poles of that function are all-important. |
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Laplace Transforms and Convolution 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. |
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Pictures of Solutions 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"". |
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Phase Plane Pictures: Source, Sink, Saddle Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exponent or eigenvalue. |
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Phase Plane Pictures: Spirals and Centers Imaginary exponents with pure oscillation provide a ""center"" in the phase plane. The point (position, velocity) travels forever around an ellipse. |
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Two First Order Equations: Stability A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular). |
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Linearization at Critical Points A critical point is a constant solution to the differential equation. The slope of the right hand side decides stability or instability. |
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Linearization of Two Nonlinear Equations With two equations, the two linearized equations use the 2 by 2 matrix of partial derivatives of the right hand sides. |
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Eigenvalues and Stability: 2 by 2 Matrix, A Two equations with a constant matrix are stable (solutions approach zero) when the trace is negative and the determinant is positive. |
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The Tumbling Box in 3-D A box in the air can rotate stably around its shortest and longest axes. Around the middle axis it tumbles wildly. |
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The Column Space of a Matrix Capturing all combinations of the columns gives the column space of the matrix. It is a subspace (such as a plane). |
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Independence, Basis, and Dimension 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. |
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The Big Picture of Linear Algebra 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. |
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Graphs 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. |
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Incidence Matrices of Graphs The incidence matrix has a row for every edge, containing -1 and +1 to show which two nodes are connected by that edge. |
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Eigenvalues and Eigenvectors 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. |
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Diagonalizing a Matrix A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λ is the eigenvalue matrix. |
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Powers of Matrices and Markov Matrices Diagonalizing a matrix also diagonalizes all its powers. |
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Solving Linear Systems An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combinations of those solutions lead to all solutions. |
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The Matrix Exponential The shortest form of the solution uses the matrix exponential multiplying the starting vector (the initial condition). |
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Similar Matrices If A and B are ""similar"" then B has the same eigenvalues as A. |
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Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues. |
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Second Order Systems An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues. |
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Positive Definite Matrices A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy. |
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Singular Value Decomposition (the SVD) 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. |
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Boundary Conditions Replace Initial Conditions A second order equation can change from two initial conditions to boundary conditions at two points. |
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Laplace Equation Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region. |
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Fourier Series A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions. |
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Examples of Fourier Series Even functions use only cosines and odd functions use only sines. The coefficients in the Fourier series come from integrals. |
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Fourier Series Solution of Laplace's Equation 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. |
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Heat Equation The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth. |
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Wave Equation 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. |
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Euler, ODE1 Instructor: Cleve Moler 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. |
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Midpoint Method, ODE2 Instructor: Cleve Moler 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. |
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Classical Runge-Kutta, ODE4 Instructor: Cleve Moler 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. |
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Order, Naming Conventions Instructor: Cleve Moler 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. |
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Estimating Error, ODE23 Instructor: Cleve Moler 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. |
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ODE45 Instructor: Cleve Moler 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. |
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Stiffness, ODE23s, ODE15s Instructor: Cleve Moler 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. |
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Systems of Equations Instructor: Cleve Moler 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. |
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The MATLAB ODE Suite Instructor: Cleve Moler The MATLAB documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite. |
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Tumbling Box Instructor: Cleve Moler 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. |
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Predator-Prey Equations Instructor: Cleve Moler 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. |
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Lorenz Attractor and Chaos Instructor: Cleve Moler 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. |