How Do You Find Coefficients for Boundary Functions in Sturm-Liouville Problems?

[Mappe]

Hello.
I have a big test tomorrow, and there is one thing I can't seem to figure out:

In Sturm-Liouville problems, when the legendre polynomials is the solution to the equation, and the boundary-conditions is a function of some sort, I am trying to find the coefficients for expressing the boundary function in terms of a sum of legendre polynomials.

But how do I find the general coefficient for every order of the legendre polynomial? I mean, how do I integrate a general legendre polynomial multiplied by a function? My specific boundary function is x^2.

Thanx and please be quick :O

 

[gabbagabbahey]

Take advantage of the orthoganality of the Legendre polynomials,

\int_{-1}^{1} P_m(x)P_n(x)dx=\frac{2}{2n+1}\delta_{mn}

Or, equivalently

\int_{0}^{\pi}P_m(\cos\theta)P_n(\cos\theta)\sin\theta d\theta=\frac{2}{2n+1}\delta_{mn}

 

[Mappe]

Thanx, but I meant to integrate the legendre with another function, <f,P(n)>, the inner product. My function is x^2, so the integral/inner product will be <x^2,P(n)>, to find the coefficients for a series-expression of x^2 with legendre as the base.

 

[gabbagabbahey]

Okay, so you are having trouble evaluating the integrals, I see.

There are at least two ways I can think of:

(1)Use the Rodriguez formula and integrate by parts (straightforward but slightly difficult)

(2)Take advantage of the fact that your function in this case is only second order, and hence will only be a sum of the first three Legendre Polynomials (P_0(x), P_1(x), and P_2(x)). Start with the highest order Polynomial (the one that contains an x^2 term and realize that in order to get 1x^2, you must multiply it by \frac{2}{3}, but \frac{2}{3}P_2(x)=x^2-\frac{1}{3} and so you must get rid of the constant term, which you can do by adding \frac{1}{3}P_0(x)...hence, x^2=\frac{2}{3}P_2(x)+\frac{1}{3}P_0(x) and you can evaluate \int_{-1}^{1}x^2P_n(x)dx by using the orthoganility relationship above.

This second method is by far the easiest for low ordered polynomials like x^2, but I recommend you also give the other method a try in case you run into a more complicated boundary function on your exam.