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

2. Motivation Conservation of mass: Steady Louisiana Tech University
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3. Tangential Annular Flow
Conservation of Momentum (r-component):
No changes with z
Louisiana Tech University
Ruston, LA 71272

4. Tangential Annular Flow
Conservation of Momentum ( -component):
Steady
No changes with z
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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.
Louisiana Tech University
Ruston, LA 71272

6. The Differential Equation
Sturm and Liouville investigated the following ordinary differential equation:
Or equivalently:
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7. Exercise Problem: If What does: reduce to? Louisiana Tech University
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8. Exercise What are the solutions to ? Louisiana Tech University
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9. Relation to Bessel FunctionsIf
Reduces to what?
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10. Relation to Bessel Functions
Is Bessel’s equation:
with solution
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11. Another Relation to Bessel Functions
If:
Also reduces to Bessel’s equation:
with solution
Louisiana Tech University
Ruston, LA 71272

12. Significance of Sturm-Liouville
The 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 University
Ruston, LA 71272

13. Series Example, Bessel
For example, the orthogonality of cosines (slides 4 and 5) allows us to write:
Which is the well-know Fourier series.
Louisiana Tech University
Ruston, LA 71272

14. Series Example, Bessel Functions
Also, the orthogonality of Bessel functions (slide 9) allows us to write:
and, the orthogonality of slide 11 allows us to write:
Note the difference. The first equation is summed over different values of l in the argument, while the second equation is summed over different orders of the Bessel function.
Louisiana Tech University
Ruston, LA 71272

15. The Boundary Conditions
Sturm and Liouville showed that if:
and if, for certain values lk of of l:
Then:
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16. Example: Cosine If: then and Because the functions
are different solutions of the differential equation that satisfy the general boundary conditions at x=0, p.
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17. The Boundary Conditions
That is, the general boundary conditions:
are satisfied for integer values of m and n if we take:
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18. Zero Value or Derivative
Exercise: If
Where is f (x) zero?
Where is its derivative zero?
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19. Visual of the Cosine m = 1 case Derivative is zero here
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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:
Provided that lm and ln are values of l for which the Bessel function is zero at x = 1.
Louisiana Tech University
Ruston, LA 71272

21. Converting To Sturm Liouville
If an equation is in the form:
Divide by P(x) and multiply by:
(Integrating Factor)
Then:
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22. Converting To Sturm LiouvilleIf
then So
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23. Converting To Sturm Liouville
Compare
to the Sturm-Liouville equation
to see that the two equations are the same if:
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Ruston, LA 71272