Louisiana Tech University Ruston, LA 71272 Slide 1 Sturm-Liouville Cylinder Steven A. Jones BIEN 501 Wednesday, June 13, 2007

Louisiana Tech UniversityRuston, LA 71272

Slide 1

Sturm-Liouville Cylinder

Steven A. Jones

BIEN 501

Wednesday, June 13, 2007


Slide 2

Motivation

0

11

z

vv

rr

rv

rtzr

vv

r0

1

0rv

Conservation of mass:

Steady 0zv

rvvei ..


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


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


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


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


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


Slide 8

Exercise

What are the solutions to

2

2

,, 0,

d xx a x b

dx

?

, cos sinx A x B x


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


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


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 ,


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.


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.


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.


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,,


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,


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


Slide 18

Zero Value or Derivative

Exercise:

If

cosf x A t

Where is f (x) zero?

Where is its derivative zero?


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


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.


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


Slide 22


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


Slide 23


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

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Louisiana Tech University Ruston, LA 71272 Slide 1 Sturm-Liouville Cylinder Steven A. Jones BIEN 501 Wednesday, June 13, 2007