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

Chapter

## Abstract

At this stage we return to the full thermoelastic equations (1.1.1) and (1.1.2), that is to say to the equations in which the inertial term These are our original boundary conditions (1.1.3).

$$ \begin{gathered} \frac{{{\partial^2}\theta }}{{\partial {x^2}}} = \frac{{\partial \theta }}{{\partial t}} + \sqrt {a} \cdot \frac{{{\partial^2}u}}{{\partial x\,\partial t'}} \hfill \\ \frac{{{\partial^2}u}}{{\partial {x^2}}} = \sqrt {a} \cdot \frac{{\partial \theta }}{{\partial x\,}} + b\frac{{{\partial^2}u}}{{\partial {t^{{2'}}}}} \hfill \\ \end{gathered} $$

*b∂*^{2}*u/∂t*^{2}is retained: we suppose, as before, that the faces of the body are clamped and, therefore, that$$ u{\left| {_{{x = 0}} = u} \right|_{{x = 0}}} = 0 $$

## Keywords

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

© Springer-Verlag New York Inc. 1985