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. The modes are sin(nx) with L equals n squared. What is the third eigenvalue?
Answer: ______________
Mia studies u_t = k u_xx and says separation gives X''/X = T'/(kT), hence X'' + L X = 0 with T' + k L T = 0. Is Mia right?
Circle one: True False
A rod of length pi has both ends held at zero temperature. The separated space modes are sin(nx) with eigenvalues L = n squared. What is the third eigenvalue?
Answer: ______________
For u_t equals u_xx, separation u equals XT gives which pair of ordinary equations?
- X double prime plus L X equals 0 with T prime plus L T equals 0
- X prime plus L X equals 0 with T double prime plus L T equals 0
- X double prime minus L X equals 0 with T prime minus L T equals 0
A rod of length 2 settles to a steady state with the left end at 0 and the right end at 20. What is the steady temperature at the midpoint?
- 0
- 5
- 20
- 10
On a rod of length pi with zero ends, the initial profile is f(x) = 2 sin x + 5 sin 3x. Its sine series is b_1 sin x + b_2 sin 2x + b_3 sin 3x + .... What is b_3?
Answer: ______________
A rod of length 2 has insulated ends, so the space modes are cos(n pi x/2) with eigenvalues L = (n pi/2) squared. What is the smallest positive eigenvalue?
- pi^2
- pi/2
- pi^2/4
- pi^2/2
The integral from 0 to pi of sin x times sin 3x equals 0 because distinct sine modes are orthogonal.
Circle one: True False
On a rod of length pi with zero ends, each heat mode decays like e to the negative Lt with L equals n squared. What is the slowest decay rate?
Answer: ______________
For X'' + L X = 0 with X(0) = X(pi) = 0, which value of L admits a nonzero solution?
- L = -4
- L = 4
- L = 0
- L = 2

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

9 · Zero ends force whole n, and the third mode has L equals 3 squared.
True · With conductivity k the split reads X''/X = T'/(kT), so her constant and her two equations are exactly right.
9 · Dirichlet ends on a rod of length pi force X = sin(nx) with L = n squared, so the third mode has L = 9.
X double prime plus L X equals 0 with T prime plus L T equals 0 · Dividing through gives X double prime over X equals T prime over T, shared as negative L.
10 · The steady profile is the straight line joining the end values, and its midpoint value is 10.
5 · Matching 2 sin x + 5 sin 3x against b_1 sin x + b_2 sin 2x + b_3 sin 3x gives b_3 = 5.
pi^2/4 · Insulated ends on a rod of length 2 give L = (n pi/2) squared, so the smallest positive one is pi^2/4.
True · Orthogonal modes integrate to zero against each other on this rod.
1 · Larger L decays faster, so the smallest L gives the slowest rate.
L = 4 · Only L = 4 gives sin(2x), which vanishes at both ends; the other values force the trivial solution.