louisiana tech university ruston, la 71272 slide 1 sturm-liouville cylinder steven a. jones bien 501...
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![Page 1: Louisiana Tech University Ruston, LA 71272 Slide 1 Sturm-Liouville Cylinder Steven A. Jones BIEN 501 Wednesday, June 13, 2007](https://reader035.vdocuments.mx/reader035/viewer/2022062803/56649f055503460f94c1a997/html5/thumbnails/1.jpg)
Louisiana Tech UniversityRuston, LA 71272
Slide 1
Sturm-Liouville Cylinder
Steven A. Jones
BIEN 501
Wednesday, June 13, 2007
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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 ..
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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
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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
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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
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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
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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
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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
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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
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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
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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 ,
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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.
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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.
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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.
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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,,
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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,
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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
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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?
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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
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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.
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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
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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
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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