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Unformatted text preview: Find the solution u s at steady state, i.e. u t → 0. • Verify that u n = en 2 π 2 t sin( nπx ) satisﬁes the PDE and the homogeneous form of the boundary conditions, u (1 ,t ) = 0, (the nπ are the eigenvalues of the homogeneous problem). • Verify that u = u s + ∑ ∞ n =1 A n u n ( x,t ) satisﬁes the nondimensional PDE, where A n are arbitrary coeﬃcients. • Show that Z 1 sin nπx sin mπx dx = ( for n 6 = m 1 2 for m = n • Deduce that the coeﬃcients A n can be computed from the initial condition u ( x,t = 0) = g ( x ) as A n = 2 Z 1 g ( x ) sin nπx d x • Find the coeﬃcients when g ( x ) = sin πx + 1 5 sin 5 πx . • Draw the transient solution at nondimensional times t = 0, 0 . 1, 0 . 5,1 and 2. What do you notice about the decays of the diﬀerent wave numbers?...
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 Spring '08
 Iskandarani,M
 Thermodynamics, Numerical Analysis, Boundary value problem, Mathematica

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