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

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

:math:`\frac{T(x,t)-{T}_{p}}{{T}_{0}-{T}_{p}}=\frac{4}{\pi }\sum _{n=1}^{\infty }\frac{1}{n}\mathrm{sin}(\frac{n\pi x}{\mathrm{2L}})\mathrm{exp}\left\{-{(\frac{n\pi }{\mathrm{2L}})}^{2}\mathrm{.}\frac{\lambda }{\rho {C}_{p}}\mathrm{.}t\right\}`


.. csv-table::

    ":math:`x=` ", "abscissa"
    ":math:`t=` ", "Time"
    ":math:`{T}_{0}=` ", "Initial temperature"
    ":math:`{T}_{p}=` ", "Imposed temperature"
    ":math:`n=` "," :math:`\mathrm{1,3}\mathrm{,5},\mathrm{...}`"



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

Temperatures at points :math:`\mathrm{M1}` (:math:`x=0.2`) and :math:`\mathrm{M2}` (:math:`x=0.8`), and at various times (:math:`t=0.1,0.2,0.7` and :math:`2.0`).

The reference values are those given in guide VPCS.


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

Digital series.


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


* J.F. SACCADURA: Introduction to heat transfers, Paris, Technique and documentation (1982).