Understanding Sturm-Liouville Equation: Applications and Solutions

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 Ruston, LA 71272

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 Louisiana Tech University Ruston, LA 71272

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: Louisiana Tech University Ruston, LA 71272

7. Exercise Problem: If What does: reduce to? Louisiana Tech University Ruston, LA 71272

8. Exercise What are the solutions to ? Louisiana Tech University Ruston, LA 71272

9. Relation to Bessel Functions If Reduces to what? Louisiana Tech University Ruston, LA 71272

10. Relation to Bessel Functions Is Bessel’s equation: with solution Louisiana Tech University Ruston, LA 71272

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: Louisiana Tech University Ruston, LA 71272

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. Louisiana Tech University Ruston, LA 71272

17. The Boundary Conditions That is, the general boundary conditions: are satisfied for integer values of m and n if we take: Louisiana Tech University Ruston, LA 71272

18. Zero Value or Derivative Exercise: If Where is f (x) zero? Where is its derivative zero? Louisiana Tech University Ruston, LA 71272

19. Visual of the Cosine m = 1 case Derivative is zero here Derivative is zero here Louisiana Tech University Ruston, LA 71272

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: Louisiana Tech University Ruston, LA 71272

22. Converting To Sturm Liouville If then So Louisiana Tech University Ruston, LA 71272

23. Converting To Sturm Liouville Compare to the Sturm-Liouville equation to see that the two equations are the same if: Louisiana Tech University Ruston, LA 71272