8  It Gets Hot, but When?

The standard technique of separation of variables and Fourier series works. Many texts explain how to use these techniques for the case that the boundary is held constant at 0 and an initial temperature is given for the interior,  and for the case where the boundary is nonconstant and the steady-state solution is sought. It is not hard to combine these two methods to solve the problem at hand. The answer is:
0.4240113870336883637974336685932564512477620906642747621971124959133101769575636922970724422944770112.
The following is a very terse solution, where some numerical work was used to deduce that the steady state solution at the center is exactly 5/4 (I discovered this numerically, but it is a special case of a known family of series (I. J. Zucker, 1984, see Problem 10), and also follows from the symmetry of the problem. Only three terms of the series were used, which additional computation shows is adequate. Using more terms yields 500 digits easily. By using the explicit partial sum in the code and simplifying the equation a bit, the final expression is nicely simple (essentially a ninth-degree polynomial equation).

Here is the series that converges to 5/4 (the result alluded to above, with a multiplicative constant stripped off, so the sum is different).
        
          

F. Bornemann observes that this series is a special case of a more general one which can be expressed in terms of the modular elliptic function (see #10).

Created by Mathematica  (June 27, 2004)