Example Problem Set 5

1.

Suppose we need to solve a linear 2nd-order differential equation with constant coefficients A and B in which the combined derivatives of the solution we seek, f(x), are only known to be related to another function, g(x):

$$A \frac{d^{2}f(x)}{dx^{2}} + B \frac{df(x)}{dx} = g(x)$$

We know the function g(x), and we can compute its Fourier Transform $G(\mu)$.

How can we use the properties of the Fourier Transform and its inverse in order to compute the solution f(x) of this differential equation? Provide an expression for $F(\mu)$, the Fourier Transform of f(x), in terms of $G(\mu)$, frequency variable $\mu$, and the coefficients in the differential equation.

What final step is now required in order to compute the actual solution f(x)of the differential equation, given your expression for $F(\mu)$?

2.

In numerical computing, differential operators must always be represented by finite differences. Assume that a function has been sampled at uniform, closely spaced, intervals. How many consecutive sample points are necessary in order to compute the N^th derivative of the function at some point?

To compute the $3^{\mbox{rd}}$ derivative in a local region of a function, what set of weights would you use to multiply consecutive samples of the function?

3.

What is the principal computational advantage of using orthogonal functions, over non-orthogonal ones, when representing a set of data as a linear combination of a universal set of basis functions?

If $\Psi_{k}(x)$ belongs to a set of orthonormal basis functions, and f(x) is a function or a set of data that we wish to represent in terms of these basis functions, what is the basic computational operation we need to perform involving $\Psi_{k}(x)$ and f(x)?