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To derive the expression for the temperature \(T(x,t)\) at a given time \(t\) in the case of transient heat conduction (also known as unsteady-state heat conduction), we typically use the heat conduction equation. The basic heat conduction equation in one dimension is given by: \[ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \] where: - \(T(x,t)\) is the temperature at position \(x\) and time \(t\), - \(\alpha = \frac{k}{\rho c}\) is the thermal diffusivity of the material, with \(k\) being the thermal conductivity, \(\rho\) the density, and \(c\) the specific heat capacity. To solve this numerically, we apply initial and boundary conditions. The typical scenario involves the following: 1. Initial Condition: The temperature of the solid at time \(t ______ ________ ____ __________ _________ _____.
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