2024 Fall Qualifying Exam in Applied Mathematics (AI-generated)

Part A

Problem A1.

For

where and , find a leading order boundary-layer approximation.

Proof.

The reduced outer problem is

Thus

The convection coefficient of is positive, so the outer solution satisfies the right boundary condition . Hence

The remaining condition at is enforced by a layer of thickness . Put and . The leading layer equation is

so . Matching gives , and gives . Therefore

Problem A2.

Use the contraction mapping theorem for with locally Lipschitz; give a continuous nonunique example; state smooth dependence if .

Proof.

Let and

On the ball , suppose and is Lipschitz with constant . If and , then maps the ball into itself and

Thus has a unique fixed point, which is exactly the IVP solution.

For nonuniqueness with continuous , take

Both and for solve the IVP. If , then the flow depends on the initial data and is as smooth in time as the equation permits.

Problem A3.

For

show existence of a periodic orbit, give Floquet multipliers in integral form, and determine stability.

Proof.

The radial equation satisfies

Thus for small and for large. Also the angular velocity is bounded away from zero in the annular trapping region. By the Poincare--Bendixson theorem, a periodic orbit exists.

Let the periodic orbit be with period . One Floquet multiplier is . The product of the multipliers equals

Here

Hence the nontrivial multiplier is

Since the orbit is attracting in the radial direction, this integral is negative; therefore , and the periodic orbit is locally asymptotically stable.

Part B

Problem B1.

Analyze the multirate explicit Euler method in the exam and apply it to , .

Proof.

The method uses two half steps for and one full step for . Taylor expansion shows the update is locally second order at integer times when is treated at the prescribed multirate level; the update is the usual explicit Euler update.

For the linear system,

Also

Therefore

The amplification matrix is triangular with eigenvalues

Both have modulus less than precisely when

Problem B2.

State the least squares normal equations, uniqueness criteria, residual uniqueness, and solve the exam's minimum-norm example.

Proof.

A least squares solution satisfies

It is unique iff , i.e. has full column rank. The residual is always unique because is the orthogonal projection of onto .

For

the exact solution set is . Minimize

The minimizer is , hence

Problem B3.

Prove equals the sum of eigenvalues, derive the relative error estimate for , and discuss ill-conditioning of

Proof.

The coefficient of in is . Since the same polynomial is , the coefficient is also . Hence .

If , then , so

Since ,

For , . One eigenvalue is near and the other is near , so . Thus it is ill-conditioned for small .

Part C

Problem C1.

Find the shortest path on the unit sphere between two points and .

Proof.

The minimizer is the shorter great-circle arc joining and . In spherical coordinates the length functional is

The Euler--Lagrange equations are the geodesic equations for the round sphere. Their solutions are great circles, because a unit-speed geodesic has acceleration normal to the sphere. The plane through intersects the unit sphere in a great circle, and the shorter arc has length

which is the spherical distance. Hence it is the global minimum.

Problem C2.

For a frictionless harmonic oscillator, write the Lagrangian, Euler--Lagrange equation, Hamiltonian system, and solve for position.

Proof.

The Lagrangian is

Euler--Lagrange gives

The momentum is , and

Hamilton's equations are

Thus

Problem C3.

Assume , the first Dirichlet eigenvalue. If solves the Euler--Lagrange equation for

prove it is a minimum.

Proof.

Let , . Taylor's theorem gives

The first variation term vanishes because satisfies the Euler--Lagrange equation. Hence

Using and Poincare's inequality,

Thus is a minimizer.