Final Examination
MEAM 501
Analytical Methods in Mechanical Engineering
1997 Fall
1.
State the following definitions, properties, and/or concept ( 40 points ) :
(1)
Define the Lagrange interpolation of a function
f
defined on an interval (
a
,
b
) by
using
n
+1 points
xx
a
b
n
11
,....
,
,
+
∈
[]
. State clearly major properties of the basis
functions of the Lagrange interpolation.
Within a given interval (a,b), we place n+1 points, say,
x
n
12
1
,
,.....
,
+
, and using the
values
ff
x
ii
=
(29
of the given function
f
(
x
) at these points, we approximate the function
f
by a
n
degree polynomial
f
n
:
fx
f
Lx
ni
i
i
n
i
j
ij
j
ji
n
(29= (29 (29 =


=
+
=
≠
+
∑
∏
1
1
1
1
,
The basis functions
i
have the following properties :
1)
they are
n
degree polynomials
and
2)
=
=
≠
1
0
if
if
(2)
State the property of the Legendre polynomilas defined on the interval (1,1) ?
They are orthogonal with respect to the inner product
fg
fxgxd
x
,
=
(29(29

∫
1
1
, and they
are obtained from the set of polynomial basis functions 1
23
, ,
,
,......
,
,.....
xx x
x
n
{}
by
GramSchmidt orthogonalization process .
(3)
Are the two functions
fx x
x
1
(29=+ (29= (29
and
sin
π
orthogonal in an interval
(0,1) ?
If not, orthogonalize them by adding an appropriate constant or polynomial to
the function
f
1
. From these make up two orthonormal basis functions.
f f
x
x dx
x
d
dx
xd
x
x
x
d
x
x
x
x
x
x
0
1
0
1
0
1
0
1
2
0
1
1
1
1
31
,
sin
cos
cos
'
cos
sin
=+
(
29=+

=
+
(
29
++
(
∫∫
∫
=
=
=
=
π
π
π
π
π
π
π
π
π
π
=
3
π
Thus they are not orthogonal We shall now consider