Consider the heat conduction in a thin insulated bar of length \(3 \mathrm{~m}\) where the initial temperature at \(t=0\) is \(f(x)=15-10 x^{\circ} \mathrm{C}\) and the ends of the bar are kept at \(0^{\circ} \mathrm{C}\). The partial differential equation for the temperature distribution \(u(x, t)\) at the distance \(x\) and time \(t\) in the bar is therefore given by \(\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\). Here the thermal diffusivity is simply equal to \(1 \mathrm{~m}^{2} \mathrm{~s}^{-1}\). The boundary conditions for this problem are given mathematically by \(u(0, t)=0, t>0, u(3, t)=0, t>0\), and the initial condition is \(u(x, 0)=15-10 x\), for \(0