## Recent **Performance Management** Questions & Answers

###### Consider the finite element method for the wave equation
utt ? uxx = 0, 0 ? x ? l, t > 0,
u (0) = u (l) = 0,
with given initial conditions.
(a) Show t...

##### Consider the finite element method for the wave equation
utt ? uxx = 0, 0 ? x ? l, t > 0,
u (0) = u (l) = 0,
with given initial conditions.
(a) Show that an appropriate requirement is that
n
i=1
A
i (t)
l
0
vi (x) vj (x) dx +
n
i=1
Ai (t)
l
0
?vi
?x
...

###### Show that the extremals of the functional
I [y (x)] =
b
a
p (x) y
2 ? q (x) y2!
dx
subject to the constraint
J [y (x)] =
b
a
r (x) y2dx = 1,
ar...

##### Show that the extremals of the functional
I [y (x)] =
b
a
p (x) y
2 ? q (x) y2!
dx
subject to the constraint
J [y (x)] =
b
a
r (x) y2dx = 1,
are solutions of the Sturm...

###### Show that the Euler...

##### Show that the Euler...

###### Find an approximate solution of the biharmonic problem
?4u = 0, R= {(x, y) : ?a < x < a, ?b < y < b}
with the boundary conditions
uxy = 0, uyy = c
1...

##### Find an approximate solution of the biharmonic problem
?4u = 0, R= {(x, y) : ?a < x < a, ?b < y < b}
with the boundary conditions
uxy = 0, uyy = c
1 ? y2
b2
for x = +a,
uxy = 0, uxx = 0 for y = +b,
where c is a constant....

###### The torsion of a prismatic rod of rectangular cross section of length 2a
and width 2b is governed by
?2u = 2 in R = {(x, y) : ?a < x < a, ?b < y < b}
...

##### The torsion of a prismatic rod of rectangular cross section of length 2a
and width 2b is governed by
?2u = 2 in R = {(x, y) : ?a < x < a, ?b < y < b}
u = 0 on ?R.
(a) Find an approximate solution u1 (x, y). Hence, calculate the torsional
moment M for a = b and for a = b.
(b) Find the exact classica...

###### Show that there are an infinite number of continuous functions with
piecewise continuous first derivatives that minimize the functional
I [y (x)] =
...

##### Show that there are an infinite number of continuous functions with
piecewise continuous first derivatives that minimize the functional
I [y (x)] =
2
0
y
2 (1 + y
)2
dx
with y (0) = 1 and y (2) = 0....

###### Derive the Euler...

##### Derive the Euler...

###### Show that the Euler...

##### Show that the Euler...

###### Show that the Euler...

##### Show that the Euler...

###### (a) If the functional I in (14.6.3) depends on two functions u and v,
that is,
I (u, v) =
b
a
F (x, u, v, u
, v
) dx,
show that there are two Eul...

##### (a) If the functional I in (14.6.3) depends on two functions u and v,
that is,
I (u, v) =
b
a
F (x, u, v, u
, v
) dx,
show that there are two Euler...