Classifying Second-Order PDEs and What the Type Decides · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Sorting second-order equations by their type

Mathematics · Differential Equations · ages 22-23
Name ______________________   Date ____________

  1. You start a parabolic equation, like the heat equation, with a sharp kink in the data. What happens to the kink as time runs?

    • It smooths out right away
    • It travels along unchanged forever
    • It grows into a bigger and bigger spike

  2. Look at the equation u_xx + 4u_yy = 0 (the second x derivative plus 4 times the second y derivative equals zero). Using the discriminant B^2 - 4AC, what type is this equation?

    • Elliptic
    • Parabolic
    • Hyperbolic

  3. The wave equation is u_tt - 9u_xx = 0 (the second t derivative minus 9 times the second x derivative equals zero). What type is it?

    • Elliptic
    • Parabolic
    • Hyperbolic

  4. The heat equation u_t equals 5u_xx has what type?

    • Elliptic
    • Hyperbolic
    • Parabolic

  5. A wave equation is set on a string. Which starting data makes it well posed?

    • The initial position and the initial velocity
    • Only the initial position
    • The temperature at the final time

  6. True or false: the type of a second-order equation (elliptic, parabolic, or hyperbolic) is decided only by the coefficients of its second-order terms, not by the first-order or zero-order terms.

    Circle one:   True   False

  7. A hyperbolic equation like the wave equation is set up on a string. Which starting data makes the problem well posed?

    • The initial position and the initial velocity of the whole string
    • Only the initial position, with no velocity given
    • The temperature of the string at the final time

  8. You have an elliptic equation like Laplace's equation on a square region. Which data gives a well-posed problem?

    • The value of u at every point of the boundary of the square
    • The value of u and its time derivative on one line at time zero
    • The value of u on just one edge, with nothing on the other edges

  9. For 3u_xx plus 2u_xy plus 3u_yy equals 0, compute B squared minus 4AC and name the type.

    • 0, parabolic
    • Negative 32, elliptic
    • 40, hyperbolic

  10. A sharp kink in the starting data of the heat equation is still visible one second later.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Sorting second-order equations by their type W1-mt_hlQWbd1kzR-s1

  1. It smooths out right away · Parabolic equations diffuse: sharp corners in the data are smoothed out instantly, and the solution becomes gentle and rounded.
  2. Elliptic · Here A = 1, B = 0, C = 4, so B^2 - 4AC = -16, which is negative. A negative discriminant means elliptic, like Laplace's equation.
  3. Hyperbolic · Treating t like one variable and x like the other, A = 1, B = 0, C = -9, so B^2 - 4AC = 36, which is positive. Positive means hyperbolic.
  4. Parabolic · The discriminant is 0, and zero means parabolic.
  5. The initial position and the initial velocity · Second order in time needs two initial conditions: position and velocity.
  6. True · The discriminant B^2 - 4AC uses only A, B, and C, the coefficients of u_xx, u_xy, and u_yy. Lower-order terms do not change the type.
  7. The initial position and the initial velocity of the whole string · Hyperbolic equations are second order in time, so they need two pieces of initial data: where things start and how fast they are moving. One piece alone leaves the solution undetermined.
  8. The value of u at every point of the boundary of the square · Elliptic equations describe steady states, and they need a boundary condition around the whole closed boundary. Cauchy-style data on one line is what suits hyperbolic equations instead.
  9. Negative 32, elliptic · A 3, B 2, C 3 gives 4 minus 36, which is negative 32: elliptic.
  10. False · Parabolic diffusion smooths every kink right away instead of carrying it.
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