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  3. 2 inhomogeneous boundary conditions consider the heat equation that governs...

Question: 2 inhomogeneous boundary conditions consider the heat equation that governs...

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