Post on 13-Jan-2016
Louisiana Tech UniversityRuston, LA 71272
Slide 1
Sturm-Liouville Cylinder
Steven A. Jones
BIEN 501
Wednesday, June 13, 2007
Louisiana Tech UniversityRuston, LA 71272
Slide 2
Motivation
0
11
z
vv
rr
rv
rtzr
vv
r0
1
0rv
Conservation of mass:
Steady 0zv
rvvei ..
Louisiana Tech UniversityRuston, LA 71272
Slide 3
Tangential Annular Flow
r
rzrrr
rz
rrr
r
fz
S
r
SS
rr
rS
rr
P
z
vv
r
vv
r
v
r
vv
t
v
11
2
0rv
Conservation of Momentum (r-component):
No changes with z
r
vf
r
SS
rr
rS
rr
Pr
rrr211
0
Louisiana Tech UniversityRuston, LA 71272
Slide 4
Tangential Annular Flow
fz
SS
rr
Sr
r
P
r
z
vv
r
vvv
r
v
r
vv
t
v
zr
zr
r
111 2
2
0rv
r
rr
r fr
Sr
r
Pf
r
Sr
r
P
r
22
2
1,
110 or
Conservation of Momentum ( -component):
No changes with z
Steady rvv 0rv rvv
rvvvv zr ,0,0
Louisiana Tech UniversityRuston, LA 71272
Slide 5
Motivation
We have seen that the orthogonality relationships, such as:
Are useful in solving boundary value problems. What other orthogonality relationships exist?
It turns out that similar relationships exist for Legendre functions, Bessel functions, and others.
nmif
nmifdxnxmx
2
0coscos
0
Louisiana Tech UniversityRuston, LA 71272
Slide 6
The Differential Equation
Sturm and Liouville investigated the following ordinary differential equation:
bxaxxwxqdx
xdxp
dx
d
,0,,
Or equivalently:
2
2
, ,, 0,
d x d xp x p x q x w x x
dx dxa x b
Louisiana Tech UniversityRuston, LA 71272
Slide 7
Exercise
bxaxxwxqdx
xdxp
dx
d
,0,,
Problem: If
What does:
1, 0, 1p x q x w x
reduce to?
bxaxdx
xd ,0,
,2
Louisiana Tech UniversityRuston, LA 71272
Slide 8
Exercise
What are the solutions to
2
2
,, 0,
d xx a x b
dx
?
, cos sinx A x B x
Louisiana Tech UniversityRuston, LA 71272
Slide 9
Relation to Bessel Functions
If
0,,
xxwxqdx
xdxp
dx
d
Reduces to what?
0,,, 22
2
22 xnx
dx
xdx
dx
xdx
2, , ,p x x q x n x w x x
Louisiana Tech UniversityRuston, LA 71272
Slide 10
Relation to Bessel Functions
0,,
xxwxqdx
xdxp
dx
d
Is Bessel’s equation:
0,,, 22
2
22 xnx
dx
xdx
dx
xdx
with solution , nx J x
Louisiana Tech UniversityRuston, LA 71272
Slide 11
Another Relation to Bessel Functions
If:
0,,
xxwxqdx
xdxp
dx
d
Also reduces to Bessel’s equation:
0,,, 222
2
22
xx
dx
xdx
dx
xdx
xxwxxqxxp 1,, 2
with solution rJx ,
Louisiana Tech UniversityRuston, LA 71272
Slide 12
Significance of Sturm-LiouvilleThe previous slides show that Sturm-Liouville is a general form that can be reduced to a wide variety of important ordinary differential equations. Thus, theorems that apply to Sturm-Liouville are widely applicable.
We will see that the orthogonality property which arises from the Sturm-Liouville equation allows us to write functions as infinite sums of the characteristic functions of an equation.
Louisiana Tech UniversityRuston, LA 71272
Slide 13
Series Example, Bessel
For example, the orthogonality of cosines (slides 4 and 5) allows us to write:
0
0
sincos
n
tin
nnnnn
neCxf
tBtAxf
or
Which is the well-know Fourier series.
Louisiana Tech UniversityRuston, LA 71272
Slide 14
Series Example, Bessel Functions
Also, the orthogonality of Bessel functions (slide 9) allows us to write:
0k
knk xJAxf
and, the orthogonality of slide 11 allows us to write:
0n
nn xJAxf
Note the difference. The first equation is summed over different values of in the argument, while the second equation is summed over different orders of the Bessel function.
Louisiana Tech UniversityRuston, LA 71272
Slide 15
The Boundary Conditions
0,,
xxwxqdx
xdxp
dx
d
and if, for certain values k of of :
bxxBdx
xdB
axxAdx
xdA
kk
kk
at
at
0,,
0,,
21
21
Then:
Sturm and Liouville showed that if:
nmdxxxxwb
a mn for0,,
Louisiana Tech UniversityRuston, LA 71272
Slide 16
Example: Cosine
0,,
2
2
xdx
xd
then 1, xw
and
If:
nmdxmxnx
nmdxxxxwb
a nm
for
for
0coscos
0
0
Because the functions nxmx cos,cosare different solutions of the differential equation that satisfy the general boundary conditions at x=0,
Louisiana Tech UniversityRuston, LA 71272
Slide 17
The Boundary Conditions
bxxBdx
xdB
axxAdx
xdA
at
at
0,,
0,,
21
21
are satisfied for integer values of m and n if we take:
That is, the general boundary conditions:
0,0,,0 22 BAba
Louisiana Tech UniversityRuston, LA 71272
Slide 18
Zero Value or Derivative
Exercise:
If
cosf x A t
Where is f (x) zero?
Where is its derivative zero?
Louisiana Tech UniversityRuston, LA 71272
Slide 19
-1.5
-1
-0.5
0
0.5
1
0 1 2 3
x
cos
( x
)Visual of the Cosine
Derivative is zero here
Derivative is zero here
m = 1 case
Louisiana Tech UniversityRuston, LA 71272
Slide 20
Application of Sturm-Liouville to Jn
From Bessel’s equation, we have w(x) = x, and the derivative is zero at x = 0, so it follows immediately that:
nmdxxJxJx
nmdxxxxw
mn
b
a nm
for
for
0
0
1
0 00
Provided that m and n are values of for which the Bessel function is zero at x = 1.
Louisiana Tech UniversityRuston, LA 71272
Slide 21
Converting To Sturm Liouville
If an equation is in the form:
2
20
d y dyP x Q x R x y
dx dx
Divide by P(x) and multiply by:
(Integrating Factor)
Then:
Q xdx
P xp x e
2
20
Q x Q x Q xdx dx dx
P x P x P xQ x R xd y dye e e y
dx P x dx P x
Louisiana Tech UniversityRuston, LA 71272
Slide 22
Converting To Sturm Liouville
So
Q xdx
P xp x e
2
20
Q x Q x Q xdx dx dx
P x P x P xQ x R xd y dye e e y
dx P x dx P x
If then
Q xdx
P xQ xp x e
P x
2
20
R xd y dyp x p x p x y
dx dx P x
Louisiana Tech UniversityRuston, LA 71272
Slide 23
Converting To Sturm Liouville
Compare
to the Sturm-Liouville equation
2
20
R xd y dyp x p x p x y
dx dx P x
2 , ,
, 0d x d x
p x p x q x w x xdx dx
to see that the two equations are the same if:
R xq x w x p x
P x