Splitting the heat equation into space and time

Mathematics · Differential Equations · ages 22-23

Name ______________________ Date ____________

A rod of length pi has both ends held at zero, with modes sin(nx) and eigenvalues L = n squared. Which n gives eigenvalue 16?
Answer: ______________
Separating u = X(x)T(t) in u_t = u_xx gives X''/X = T'/T. Why must both sides equal the same constant?
- Because the equation is linear in u
- A function of x alone equals a function of t alone
- Because X and T must take equal values everywhere
- Because both ends of the rod are held at zero
For u_t equals 4 u_xx, separation u equals XT gives which pair of ordinary equations?
- X double prime plus L X equals 0 with T prime plus 4 L T equals 0
- X double prime plus 4 L X equals 0 with T prime plus 4 L T equals 0
- X double prime minus L X equals 0 with T prime plus 4 L T equals 0
For u_t equals u_xx, a separated product XT with X double prime plus L X equals 0 has time factor T equals e to the Lt.
Circle one: True False
On a rod of length pi with zero ends, the initial profile is f(x) = 7 sin 2x + sin 4x. What is b_2, the coefficient of sin 2x?
Answer: ______________
A rod with insulated ends has a mode with L = 0: a constant profile that never decays. Is that right?
Circle one: True False
A rod of length 4 has insulated ends, so the modes are cos(n pi x/4) with eigenvalues L = (n pi/4) squared. What is the smallest positive eigenvalue?
- pi^2/16
- pi^2/4
- pi/4
- pi^2/8
The integral from 0 to pi of sin 2x times sin 5x equals 0 because distinct sine modes are orthogonal.
Circle one: True False
A rod of length 3 has both ends held at zero, so its modes are sin(n pi x over 3) with L equals (n pi over 3) squared. Each mode decays like e to the negative Lt. What is the slowest decay rate, to two decimal places?
Answer: ______________
For X'' + L X = 0 with X(0) = X(pi) = 0, which value of L gives only the zero solution?
- L = 1
- L = 4
- L = 16
- L = 5

Answer key

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

Splitting the heat equation into space and time W1-mt_XnPDL0X1TL-s1

4 · L = n squared = 16 gives n = 4, the mode sin 4x.
A function of x alone equals a function of t alone · The left side depends only on x and the right only on t. If they are equal for every x and t, neither can change, so both equal one constant.
X double prime plus L X equals 0 with T prime plus 4 L T equals 0 · Dividing by 4XT gives X double prime over X equals T prime over 4T, shared as negative L, so the 4 sits on the time side only.
False · The time equation is T prime plus L T equals 0, whose solution decays: T equals e to the negative Lt.
7 · Distinct sine modes are orthogonal, so the coefficient of sin 2x is read off: 7.
True · Zero slope at both ends allows the constant function, with L = 0, and e^(0t) = 1 never decays.
pi^2/16 · n = 0 gives L = 0; the smallest positive value is at n = 1: (pi/4)² = pi²/16.
True · Distinct sine modes on the pi rod integrate to zero against each other.
1.1 · The smallest eigenvalue is n equals 1: (pi over 3) squared = pi squared over 9, about 1.0966, which is 1.10 to two decimal places.
L = 5 · Nonzero solutions need L = n squared. 5 is not a perfect square, so sin(√5 x) does not vanish at pi and only X = 0 fits.