10.19.2016
Dear Math 488 Student:I will be out of the country and out of email and phone contact from Friday, Oct. 21, until Monday, Oct. 31. Consequently, I have prepared the Math 488 Week 10 message in advance and have asked Peg Pisel to distribute that message to you.
I hope that your work is going well in our course and in your engineering work. Our class schedule specifies that you should complete Lectures 25, 26, and 27 and the associated homework problems by Sunday, October 30.
The group of lectures 22 through 27 constitutes a very coherent unit on power series methods for solving second order linear differential equations with variable coefficients. Such equations include Bessel's equation, Legendre's equation, and many others that have important applications in engineering. Such applications will be the main topic of Lectures 28 through 33.
You will recall that Lectures 22 and 23 presented the power series method for solving differential equations with variable coefficients and applied this method to find solutions of Airy's Equation, all Euler-Cauchy equations and Legendre's differential equation. Lectures 24 and 25 do the corresponding solution work for Bessel's equation. Bessel’s Equation is probably the most important variable coefficient linear differential equation in applied mathematics because so many other linear differential equations can be reduced to a form of Bessel's equation by a suitable change of variables.
WEEK 10 - Content Summary for Lectures 25, 26, and 27
Lecture 25 introduces Sturm-Liouville systems, which can be defined roughly in the following scary way:
A Sturm-Liouville system is a homogeneous second order linear differential equations of the form:
(S-L) d/dt[r(t) y'(t) ] + [ q(t) + Lambda p(t)] y(t) = 0
where Lambda is a non-negative real number and r(t), q(t) and p(t) are real functions defined on some interval {a, b] on the non-negative t-axis such that p(t) is non-negative on [a, b].
A second order differential equation of this type is called Sturm-Liouville Equation. Sturm-Liouville differential equations include nearly all of the second-order linear differential equations with variable coefficients that are important in engineering and applied mathematics, including all of the equations discussed in Lectures 22, 23 and 24. Thus, Sturm-Liouville equations allow us to give a unified approach to the power series solutions to all of these equations and many others.
Up to this point in the course, we have found solutions to linear differential equations that satisfied prescribed initial conditions. We called such problems initial value problems. Because Sturm-Liouville differential equations are homogeneous, they always have the zero solution y(t) =0 for all t with a < t < b. Because this solution is always there and not useful, it is called the trivial solution. Some important applications require finding all values of Lambda for which a Sturm-Liouville differential equation has non-trivial solutions. These values of Lambda are called eigenvalues of the Sturm-Liouville system, and the corresponding solutions y(t) are called eigenfunctions of the Sturm-Liouville system. The points a and b are called boundary points and the problems are called boundary value problems.
It turns out that rather general types of functions can be represented as finite or infinite series of the eigenfunctions of Sturm-Liouville problems. We will show that important orthogonal series such as Fourier, Legendre and Bessel series, among others, are all special cases of such systems. In fact, once you identify a given second order linear differential equation with boundary conditions as a Sturm-Liouville system, the properties of such systems provide much useful information about corresponding orthogonal series. We will illustrate this over and over again in Lectures 26 and 27.
This is tough content because finding the eigenvalues of each individual Sturm-Liouville problem has many individual steps, each with a slightly different twist. The good news is that the steps themselves are the same in each application. Thus, the best thing to do is to plow ahead from one lecture to the next until you see the general pattern.
I hope that you will have a productive week during my absence. I will do my best to answer any questions you may have when I return!
Tony Peressini
Have a good week,
Tony Peressini
Cell PH: 217 840 2871
anthonyperessini@gmail.com