CALCULUS OF VARIATIONS

PROBLEM

Consider the functional

depending on another function f. Our goal is to minimize T(f) where f varies in the class of functions with f(0)=0 and f(1)=1.

In M243 we have learned to optimize functions depending on variables rather than on functions. We modify our problem so that the M243 tools are applicable. Instead of allowing f to vary over all functions with f(0)=0 and f(1)=1, we shall restrict our attention to functions which are polynomials or piecewise linear. So we will not have the exact solution of the PROBLEM but hopefully we will have a reasonable approximation of the exact solution.

1. Construct a function f(x) with f(0)=0, f(1)=1, and compute T(f) for this f. Try this for other functions f with f(0)=0 and f(1)=1.
2. Find the form of f(x), the most general polynomial of degree 3, with f(0)=0 and f(1)=1.
3. Minimize T(f) over all polynomials of degree 3 with f(0)=0 and f(1)=1. Hint: Use the answer to (2) and compute T(f) using Maple. Then the problem reduces to a minimization of a function of two variables.
4. The exact solution for the main problem is the function (determined using another technique). We compare the exact solution with the approximate solution obtained in (3). One may compare by
   • Drawing the graphs of the exact and the approximate solution on [0,1]
   • Or better if we estimate the percentage error i.e by drawing the graph of
     
     for x in [0,1]. (Choose the appropriate x and y scales in the graph). Watch for trouble at points where exact(x) is zero.
5. Let f(x) be the function such that the graph of y=f(x) consists of the line segments joining the points (0,0), (1/3,2), (2/3,-1), (1,1). Draw a sketch of the graph of this function then compute T(f). Note that f' does not exist at the points so the integrals have to be computed over each subinterval separately and then added.
6. Let f(x) be the function such that the graph of y=f(x) consists of the line segments joining the points . What is f(x) on the intervals .
7. Minimize T(f) where f varies over functions of the form given in (6). i.e. Minimize as a,b vary.
8. We now have a new candidate for the approximate solution. Draw the graph of the percentage error for this new approximate solution as done in (4). Of the two approximate solutions we have computed (in (3) and (7)), which one is more accurate.
9. What could be done to obtain a more accurate approximate solution than the ones obtained in (3) or (7)? What problems do you foresee in implementing your suggestion for improved accuracy? What problems do you foresee in the application of the techniques in (3) and (7) to the brachistochrone problem?

Turn in a handwritten summary of your work on (2),(3),(4),(6),(7),(8) and (9). Also prepare a Maple report showing your work for the solution of (3),(4),(7) and (8).

 
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