MATH2115/2194 Assignment 1
In this assignment the boldface n is a parameter based on the last digit of
your student number.
n is obtained by taking the last digit in your student number and adding 2.
Eg. If your student number is 3216385, then take the last digit 5 and add 2
to get n=5+2.
1. The temperature u(x; t) within a particular rod of length n with insulated ends is
governed by the following boundary value problem
@u
@t = n
@2u
@x2 ;
@u(0; t)
@x = 0 =
@u(n; t)
@x and u(x; 0) = ( x; 0; 0 1 < x < < x < 1 n
Use the separation of variables method with u(x; t) = X(x) T(t) to determine the
temperature along the rod for t > 0.
(7 marks)
2. The vibrations along a particular beam u(x; t) with simply supported ends is governed
by the following boundary value problem,
@2u
@t2 + n2 @@x4u4 = 0;
u(0; t) = 0 = @2u(0; t)
@x2
u(n; t) = 0 = @2u(n; t)
@x2
u(x; 0) = f(x) @u(x; 0)
@t = g(x)
Use the separation of variables method with u(x; t) = X(x)T(t) to find an expression
for the eigenvlaues λn.
(8 marks)
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