This exercise considers computing the curve that produces a surface of revolution with minimum area. Given two points (
) and (
) in the plane, with
and
and
positive, assume that
(
) is a smooth, positive, function connecting these points. If
() is rotated about the
axis, the resulting surface has area
It should be noted that the curve is required to satisfy
(
) =
and
() =
. Also, unlike the brachistochrone problem,
() does not have a singularity(at either end).(at either end).
(a) Suppose the grid points are
0
= 0,
1
=
,
2
= 2, ··· ,
n+1
= 1, where
= 1/(
+ 1). Writing
write down the composite trapezoidal approximation for
.
(b) Use a centered second-order approximation for
’ at
1,
2, ··· ,
n
, and a first-order approximation for
’ at
0
and
n+1. Using these with the result from part (a), what is the resulting approximation for
?
(c) The minimum of
from part (b) can be found using the MATLAB command
where
is the approximation from part (b) and
Y
is an
-vector containing a starting guess for (
1,
2, ··· ,
n
)
T
. What would be a good, simple, and nonzero choice for
Y, and why is it a good choice?
(d) Taking
= 0,
= 1,
= 1, and
= 3, plot the numerical solution for
() for
= 4,
= 9, and
= 19.
(e) In calculus it is shown that the curve producing the minimum area is
= (1/
)cosh(
+
), where
and
are determined from the requirements that
() =
and
() =
. Determine the nonlinear equations that must be solved to find
and
. Discuss the numerical difficulties associated with solving these equations, compared to the direct solution in parts (a)–(c).
Note: The minimal surface problem has some interesting mathematical complications, and for more about this see Oprea [2007].