Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

Semi-analytical solution involving error functions:

:math:`T(x,t)=2\left\{\sqrt{\left[1+2\sqrt{(t/\pi )}\mathrm{exp}(\frac{-{x}^{2}}{\mathrm{4t}})+\mathrm{x.erfc}(\frac{x}{2\sqrt{(t)}})\right]}-1\right\}`

with :math:`\mathrm{erfc}(x)=\frac{2}{\pi }{\int }_{x}^{\infty }{e}^{-{t}^{2}}\mathrm{dt}`

Where :math:`x=` is the abscissa

:math:`t=` time

This formula is only valid for :math:`\lambda (T)=\rho c(T)=1.+\mathrm{0.5T}`


Benchmark results
----------------------

Temperature at points :math:`A` (:math:`x=0`) and :math:`E` (:math:`x=1`) at the following moments :math:`t`: :math:`t=0.1,0.3,0.5,0.7` and :math:`\mathrm{1s}`


Uncertainty about the solution
---------------------------

Unknown, due to evaluation of error functions.


Bibliographical references
---------------------------


* Segal, N. Praagman, "A fast implementation of explicit time stepping algorithms with the finite element method for a class of nonlinear evolution problems", Int. J. num. Meth. Engng, vol. 23, pp. 155-168, 1986.