**TEST 2 MA-4451**

**SOLUTIONS**

**1.**We look for a solution in the form:

where *h* satisfies:

or equivalently:

By integrating twice:

Imposing h to be bounded for r = 0 and using the boundary condition, we determine h to be:

The problem is thus reduced to solving a homogeneous PDE for v:

**2.** See notes.

**3.a.** The problem reduces to:

**b.** The boundary condition becomes:

**c.** The equation in R is a Bessel equation of order 0. The general
solution is:

Since the solution has to be bounded at 0, we obtain that B = 0. From the boundary condition we get:

from where we determine and thus the eigenvalues :

with solutions of .

Once the eigenvalues are determined we can find T:

**d.** The solution is:

In order to satisfy the initial condition we have to determine the coefficients such that:

From the formula for the coefficients (see table 4):

From the properties of Bessel functions (see table 2) we can compute the above integral. Finally:

Thu Oct 3 09:05:00 EDT 1996