Approximation by Way of the Heat Equation or the Integro-differential Equation

  • William Alan Day
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 30)


At this stage we return to the full thermoelastic equations (1.1.1) and (1.1.2), that is to say to the equations
$$ \begin{gathered} \frac{{{\partial^2}\theta }}{{\partial {x^2}}} = \frac{{\partial \theta }}{{\partial t}} + \sqrt {a} \cdot \frac{{{\partial^2}u}}{{\partial x\,\partial t'}} \hfill \\ \frac{{{\partial^2}u}}{{\partial {x^2}}} = \sqrt {a} \cdot \frac{{\partial \theta }}{{\partial x\,}} + b\frac{{{\partial^2}u}}{{\partial {t^{{2'}}}}} \hfill \\ \end{gathered} $$
in which the inertial term b∂2u/∂t2 is retained: we suppose, as before, that the faces of the body are clamped and, therefore, that
$$ u{\left| {_{{x = 0}} = u} \right|_{{x = 0}}} = 0 $$
These are our original boundary conditions (1.1.3).


Heat Flux Heat Equation Differential Inequality Equilibrium Boundary Recurrence Property 
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Copyright information

© Springer-Verlag New York Inc. 1985

Authors and Affiliations

  • William Alan Day
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland

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