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Inhomogeneous PDEby Lacero
Tags: inhomogeneous 
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#1
Apr708, 08:37 PM

P: 32

Hi all,
1. The problem statement, all variables and given/known data Determine the equilibrium temperature distribution (if it exists). For what values of B, are there solutions. 2. Relevant equations a) Ut = Uxx + 1, U(x,0) = f(x), Ux(0,t) = 1, U(L,t) = B b) Ut = Uxx + X  B, U(x,0) = f(x), Ux(0,t) = 0, U(L,t) = 0 3. The attempt at a solution a) Assume solution U(x,t) = V(x,t) + K => Ut = Vt => Uxx = Vxx  Sub into Ut = Uxx + 1 Vt = Vxx + 1 Now to get homogeneous boundary conditions, U(0,t) = V(0,t) + K = 1 ? but K = 1 => V(0,t) = 1? // Trouble at the BC b) Assume solution U(x,t) = V(x,t) + W(x) => Ut = Vt => Uxx = Vxx + W''  Sub into Ut = Uxx + Q(x) Vt = Vxx + W'' + Q Now, let W'' + Q = 0 or W'' = Q => Vt = Vxx V(0,t) = 0 V(L,t) = 0 U(x,0) = V(x,0) + W(x) = f(x) => V(x,0) = f(x)  W(x) Now Solve transient solution, v(x,t) . . . . . . V(x,t) = (2/L)sum(n=1 to inf)(int((f(x)W(x))sin(npi/L)xdx)... etc Now Solve steady state solution, W" = Q B  X W" + X + B = 0 . . . . W(x) = Scos(zx) + Tsin(zx) + Yp... etc Now U(x,t) = W(x) + V(x,t) Am I right at all??? 


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