2. (30 points) The numerical algorithms you learned in this class can be used to solve all kinds of prob- lems. In this problem, I will walk you through solving a Partial Differential Equation by transforming it into a linear system and use the LU decomposition to solve it. The physical problem we want to solve is the following. Given a metal rod of length 1, we denote by u(x,t) its temperature at the point 0 0. We keep the rod's endpoints at constant zero temperature at all times and at time t = 0 we have that u(x, t) = sin(Tu). We want to know how temperature changes on the rod with t. The partial differential equation (PDE) that describes mathematically this phenomenon is ut(x, t) = Upx(x,t), O SI 0 with boundary conditions u(0,t) = u(1,t) = 0 for all t > 0 and initial condition u(x,0) = sin(T2), 0 0. We keep the rod's endpoints at constant zero temperature at all times and at time t = 0 we have that u(x, t) = sin(Tu). We want to know how temperature changes on the rod with t. The partial differential equation (PDE) that describes mathematically this phenomenon is ut(x, t) = Upx(x,t), O SI 0 with boundary conditions u(0,t) = u(1,t) = 0 for all t > 0 and initial condition u(x,0) = sin(T2), 0