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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 tex2html_wrap_inline42 is the Bessel function of first kind and order tex2html_wrap_inline44 , 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)


Bogdan Vernescu
Mon Sep 30 09:42:41 EDT 1996