PHYCS 3730/6720 Lab Exercise


Note: If you are already familiar with Matlab, you may use it to do this exercise instead of Maple. Please be sure to show the Matlab commands you use to complete the exercises.

Exercise 1.

Bessel functions appear commonly in problems with cylindrical geometry. For example the radial profile of the normal vibrational modes of a drumhead is described by Bessel functions. Bessel functions come with an integer index Jn(x) where n = 0, 1, .... In Maple notation they are BesselJ(n,x) and in Matlab, besselj(n,x).

Use Maple to plot the Bessel function J1(x) for a range sufficient to show the first three positive roots (not counting the one at x = 0). In the answer file Mylab06.txt, give an estimate of the zeros, based on reading the plot. Just to be clear, J1(x) is known to Maple as BesselJ(1,x) and in Matlab as besselj(1,x)

Exercise 2.

Use Maple to develop the Taylor/MacLaurin series representation of the first order Bessel function J1(x) up to and including the term in x^9. Note, the O means "order of". So O(x11) means the rest of the series vanishes with x as x11. For help with the Taylor/MacLaurin series in Maple, use ?taylor. The Jm(x) Bessel functions are called BesselJ(m,x). Use ?BesselJ to learn more and see how to get related Bessel functions.

When you are satisfied with your result, copy your answer from the Maple window to the file Mylab06.txt. Please provide both input and output lines for the results you want to hand in.

The simplest way to do this is to use your mouse to copy from the Maple window and paste it into your emacs window.

Exercise 3.

With Maple use the Newton-Raphson method to find the zero of the J1(x) Bessel function near x = 4 to a tolerance of 10-4.

Hint: For this exercise you need to evaluate the derivative of the Bessel function. Maple insists on being much more precise about this procedure than most scientists are used to. First you have to find the function that is the derivative of the Bessel function. Then you have to evaluate the derivative function at the desired point. There are two ways to do this in Maple, namely, with diff or D for the derivative.

    f := x -> BesselJ(1,x);  g := x -> evalf( D(f)(x) );
    f := x -> BesselJ(1,x); g := x -> evalf( subs(y=x, diff(BesselJ(1,y),y) ) );
You don't need to write a Maple procedure for this. Use the same technique you used for the fixed point method, i.e. write an assignment statement that gives the new Newton Raphson estimate of x based on the old estimate. Then repeat by hand until you are satisfied that it has converged.

In the answer file, show the Maple commands you used and the result you found.

Exercise 4.

Use the Maple fsolve command to find the same root. Copy the command and your answer to the answer file.

Note that with fsolve, you can specify a range of x values in which to search for the root. Check it out with ?fsolve.