Mathematical and Computational Methods for Engineers
E155B, Spring 2002
Handout #2
1. A rectangular plate is bounded by x  0 , x  a , y  0 ,
and y  b . The temperatures at the boundaries are
specified as follows:
T ( x ,0 )  0
T ( a, y )  0
T (0, y )  0
T ( x, b)  T0
y
b
0
a
x
Use the method of separation of variables to determine
the steady-state temperature distribution T ( x, y )
2. A rectangular plate with two insulated sides is bounded
by x  0 , x  a , y  0 , and y  b . The boundary
conditions are specified as follows:
T ( x ,0 )  0
T
0
x x a , y
T
0
x x 0, y
T ( x, b )  T ( x )
y
b
0
a
x
Use the method of separation of variables to determine
the steady-state temperature distribution T ( x, y )
3. Consider a rod of length L and diffusivity  2 with the
following initial and boundary conditions:
T (0, t )  0
T ( L, t )  0
T ( x,0)  f ( x)
x
L
0
Use the method of separation of variables to determine
the time-dependent temperature distribution T ( x, t )
4. A rectangular plate is bounded by x  0 , x  a , y  0 ,
and y  b . The temperatures at the boundaries are
specified as follows:
T ( x ,0 )  0
T ( a, y )  0
T (0, y )  0
T ( x, b )  0
y
b
0
a
x
At t  0 the temperature distribution is given by
T ( x, y ) . Use the method of separation of variables with
the double Fourier series to determine the timedependent temperature distribution T ( x, y, t )
5. Consider a rod of length L and diffusivity  2 with nonhomogeneous boundary conditions:
T (0, t )  T1
T ( L, t )  T2
x
L
0
T ( x,0)  f ( x)
The initial temperature distribution is given by
T ( x,0)  f ( x) . Use the method of separation of
variables to determine the stead-state solution Ts (x) and
the time-dependent temperature distribution T ( x, t ) .
6. A wire of length L, cross-sectional area A, and electrical
conductivity  is internally heated by passing an
electrical current I. Determine the time-dependent
temperature distribution in the wire T ( x, t ) subject to
the following initial and boundary conditions:
T (0, t )  0
T ( L, t )  0
T ( x,0)  f ( x)
0
I
L
x
7. A string of length l is fixed at the two end points such
that u (0, t )  u (l , t )  0 . At t  0 the string is displaced
and released from its equilibrium position such
u
u ( x,0)  f ( x) and
 g ( x) . Determine the
t x ,t 0
displacement of the string as a function of time u ( x, t ) .
8. A rectangular membrane bounded by x  0 , x  a ,
y  0 , and y  b is fixed at the boundaries. At t  0
the string is displaced and released from its equilibrium
u ( x, y,0)  f ( x, y )
position
such
that
and
u
 g ( x, y ) . Determine the displacement of the
t x , y ,t 0
membrane as a function of time u ( x, y , t ) .
9. The displacement of a vibrating string is governed by
the following inhomogeneous partial differential
equation:
c2
2 y 2 y
 2  F ( x, t )
x 2
t
Solve the above PDE subject to the following initial
and boundary conditions:
y (0, t )  y ( L, t )  0
y ( x,0)  f ( x)
y
t
 g ( x)
x ,t  0