(Solved): Let an aluminum rod of length \( 50 \mathrm{~cm} \) be initially at the uniform temperature of \( ...

Let an aluminum rod of length \( 50 \mathrm{~cm} \) be initially at the uniform temperature of \( 4^{\circ} \mathrm{C} \) . Suppose that at time \( t=0 \), the end \( x=0 \) is cooled to \( -20^{\circ} \mathrm{C} \) while the end is heated to \( 11^{\circ} \mathrm{C} \), and both are thereafter maintained at those temperatures. If we want to write the problem as a nonhomogeneous heat conduction equation, then fill the boxes with appropriate numbers : \[ \begin{array}{l} \alpha^{2} u_{x x}=u_{t} \\ u(0, t)= \\ u(\quad, t)= \\ u(x, 0)= \\ u(x, t)=v(x)+w(x, t) \text { where } v(x)= \end{array} \] Please answer all parts of the question.