PROBLEM 1: 1D heat equation with non-homogeneous BCs and a heat source The temperature distribution u(x, t) in a one-dimensional wire of length L satisfies the 1-D heat equation: ∂u ∂t = k ∂ 2u ∂x2 + u 0 ≤ x ≤ L. (1) The temperature of the wire is prescribed at both ends: u(0, t) = 0, u(L, t) = TL. (2) The initial temperature distribution inside the wire is u(x, 0) = f(x). (a) Solve for the steady distribution us(x). (b) We define a new function ˜u(x, t) = u(x, t) − us(x). What is the partial differential equation that governs the evolution of ˜u(x, t)? What are the boundary conditions and initial condition satisfied by ˜u(x, t)? (c) We use separation of variables to solve for ˜u(x, t). First, look for solutions of the form ˜u(x, t) = φ(x)G(t). What are the equations satisfied by the functions φ(x) and G(t)? (d) Solve for G(t). (e) What are the boundary conditions on the function φ(x)? Solve for φ(x). (f) Express the general solution for ˜u(x, t) as an infinite series in terms of an infinite number of unknown coefficients that depend on the initial temperature profile. (g) Solve for u(x, t) for an initial temperature distribution u(x, 0) = f(x) = T0.