Green's Functions for Boundary Value Problems · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Solve once for a pinpoint, reuse for everything

Mathematics · Differential Equations · ages 21-22
Name ______________________   Date ____________

  1. With forcing one, the two halves of the solution integral at one half each give one over sixteen. What is the total as a decimal?

    Answer: ______________

  2. For minus u double prime equal f with u(0) and u(1) both zero, which formula gives the Green's function?

    • s times (1 minus x) for s below x, x times (1 minus s) above
    • One everywhere
    • x plus s everywhere
    • Zero everywhere

  3. For minus u'' = f with u(0) and u(1) both zero, which formula gives the Green's function?

    • One everywhere
    • s times (1 minus x) on one side, x times (1 minus s) on the other
    • Zero everywhere

  4. How is the Green's function built from homogeneous solutions?

    • The left solution obeying the left condition glued to the right solution obeying the right condition
    • Adding the two boundary values together
    • Integrating the forcing twice and ignoring the conditions

  5. The jump condition says right derivative minus left derivative is minus one. If the left derivative is three, what is the right derivative?

    Answer: ______________

  6. The jump says right derivative minus left derivative is minus one. If the left derivative is three, what is the right derivative?

    Answer: ______________

  7. How is the Green's function built from homogeneous solutions?

    • Add the two boundary values together
    • Multiply the forcing by x
    • The left solution obeying the left condition glued to the right solution obeying the right condition, scaled for the jump
    • Integrate the forcing twice and ignore the conditions

  8. Different boundary conditions give different Green's functions for the same operator. True or false?

    Circle one:   True   False

  9. Different boundary conditions give different Green's functions for the same operator.

    Circle one:   True   False

  10. Where does the derivative jump come from?

    • The Green's function is discontinuous
    • The forcing function is smooth
    • The boundary conditions force a kink
    • Integrating minus u double prime equal a point source across s leaves a unit drop in slope
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Answer key

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Solve once for a pinpoint, reuse for everything W1-mt_2ctFrfKIgW-s1

  1. 0.125 · One over sixteen plus one over sixteen is one over eight, which is 0.125.
  2. s times (1 minus x) for s below x, x times (1 minus s) above · Straight lines through the boundary zeros glue continuously at s with the right slope drop.
  3. s times (1 minus x) on one side, x times (1 minus s) on the other · Straight lines through the boundary zeros glue continuously at s.
  4. The left solution obeying the left condition glued to the right solution obeying the right condition · Each half meets its own end condition, then scaling sets the jump.
  5. 2 · Three minus one is two.
  6. 2 · Three minus one is two.
  7. The left solution obeying the left condition glued to the right solution obeying the right condition, scaled for the jump · Each half satisfies the equation away from s and its own boundary condition; scaling sets the jump.
  8. True · True. The halves are chosen by the conditions, so changing them rebuilds G.
  9. True · The halves are chosen by the conditions, so changing them rebuilds G.
  10. Integrating minus u double prime equal a point source across s leaves a unit drop in slope · The delta integrates to a finite step in the first derivative while u itself stays continuous.
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