Thursday, November 15, 2012

Differential Equations - Sturm-Liouville Form

The following will be on transforming an equation into Sturm-Liouville form which is useful in finding the eigenvalues/eigenfunctions of a differential equation within a given boundary. It does this by creating a Hermitian operator.

  • Operator - a series of "operations" that act upon a function, sort of like a set of instructions. This can be anything from adding and subtracting certain parts of the function to taking the derivative, or integral. This maps a function from one vector space to another, and is not necessarily one to one.
  • Hermitian Operator - an operator that is self adjoint, which means that if you put the operator into matrix form to act upon an equation, that matrix is the same when you take the conjugate transpose of it.
  • Conjugate Transpose - Taking the transpose of a matrix, which means to switch the indices from (1,2) to (2,1) for the entire matrix. Then conjugate any terms that have imaginary components, which means switch the signs of the imaginary terms.
Sturm Liouville Form
Given a second order differential operator where D2, and D represent partial derivatives acting on some variable
We will first have this operator act on the following equation where the operator takes the place of the second order derivative acting on y where lambda is an eigenvalue, explained in a separate section.
giving us the following
We can then rearrange this by dividing through by a2(x)
Now we multiply through by e to the integral of a1 over a2 and replace the operator with just y'' and y' for simplicity
Now we are going to once again regroup this into
by taking advantage of the product rule for taking derivatives, which makes the above two parts equivalent. Now we are going to define the parts of this equation by functions r, q, and p.
Then we plug into this form.

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