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EGR 511
NUMERICAL METHODS
_______________________
LAST NAME, FIRST
Problem set #9
1. Use the Galerkin Finite-Element Method to approximate the solution to the boundary-value
problem
2
2

x,
for 0  x  1 with y(0) = y(1) = 0
4
4
16
using x0 = 0, x1 = 0.3, x2 = 0.7, x3 = 1 and compare the results to the actual solution
2
1
1



y(x) = - cos x sin x + cos x.
2
2
4
3
3
6
y” +
y=
cos
2. Show that the finite difference discretization of
(x + 1)uxx + (y2 + 1)uyy - u = 1
0  x  1, 0  y  1, x = y = 1/3
with u(0,y) = y, u(1,y) = y2, u(x,0) = 0, u(x,1) = 1
is given by
10
4
1




0 
 
 5
9
3
u11   3 

 13 17

20
4
0
  u   

3
3   12  =  9 
 9
2
5
17
10

0
   u21   
 27 
 3
3
9 u

5
13 19   22   56 


 
 0

 27 
3
9
3 

3. Solve the one-dimensional heat conduction equation
d 2T
=  f(x)
dx 2
for a 10-cm rod with boundary conditions of T(0, t) = 50 and T(10, t) = 100 and a uniform heat
source of f(x) = 20. Use the trial function T = ax2 + bx + c
4. a) Use the Rayleigh-Ritz method to approximate the solution of
y” = 3x + 1,
y(0) = 0,
y(1) = 0,
using a quadratic in x as the approximating function.
b) Solve the problem by collocation, setting the residual to zero at x = 0.5.
c) Solve the problem by Galerkin’s method.
5. Develop the elements equations for a 10-cm rod with boundary conditions of T(0, t) = 40 and
T(10, t) = 100 and a uniform heat source of f(x) = 20. Employ four equal-size elements of length
= 2.5 cm. Compute the temperature distribution for the entire rod.
dc
d 2c
6 Use Galerkin’s method to develop an element equation for D 2  U - kc = 0.
dx
dx