Heat Equation

Let us imagine that a bar of length λ₀ is placed along the λ-axis, the abcissa of the left end of the bar being λ = 0 and the right end λ = λ₀. Let k denote the heat conductivity, c the specific heat, and ρ the mass density of the bar. Furthermore, let the lateral surface of the bar be perfectly insulated so that heat can flow in and flow out only through the ends of the bar. If we denote by z(λ,t) the temperature at the point of the bar at abcissa λ at the instant t, then the heat equation in the bar is 41.1$$ {z_{\lambda \lambda }}\left( {\lambda ,t} \right) = {\alpha ^2}{z_t}\left( {\lambda ,t} \right) \left( {\alpha = \sqrt {c\rho /k} } \right). $$