# Question: inhomogeneous boundary conditions consider the heat equation that governs a...

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Inhomogeneous boundary conditions. Consider the heat equation that governs a thin rod. The temperature u(x, t) in the rod obeys ∂u ∂t = k ∂ 2u ∂x2 (6) u(0, t) = 0 (7) u(L, t) = u2 (8) u(x, 0) = f(x) (9) where the k, the thermal diffusivity, is constant. (a) Is the PDE (6) homogeneous? Are the boundary conditions (7)-(8) homogeneous? (b) Find the steady state solution of (6)-(9), uss(x). (c) Make the dependent variable transformation v(x, t) = u(x, t) − uss(x) and show that v(x, t) obeys ∂v ∂t = k ∂ 2 v ∂x2 (10) v(0, t) = 0 (11) v(L, t) = 0 (12) v(x, 0) = f(x) − uss(x) (13) (d) Find v(x, t) using separation of variables. Leave the expansion coefficients in integral (or inner product) form. (e) What is the steady state limit of v(x, t)? What is the steady state limit of u(x, t)?