**TEST 2 MA-4451**

**1.** The equation of a damped circular membrane is given by
the non-homogeneous partial differential equation:

boundary condition *u*(2,*t*) = 0 and initial condition *u*(*r*,0) = *g*(*r*).
Reduce the problem to a homogeneous problem. (20 pts)

**2.** If is the Bessel function of first kind and order ,
prove the following formulas: (10 pts)

**3.** Consider the boundary value problem:

a. By choosing *u*(*r*,*t*) = *R*(*r*) *T*(*t*) reduce the problem to two
ordinary differential equations for *R* and *T*. (20pts)

b. What are the appropriate conditions for the solutions of the above ordinary differential equations? (5 pts)

c. Find the eigenvalues and eigenfunctions for the ordinary differential equations. (15pts)

d. Write the solution of the boundary value problem in the form of a Fourier-Bessel series. (10pts)

e. Determine the coefficients of the above series. (20pts)

Mon Sep 30 09:42:41 EDT 1996