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# pb1_10 - Find the solution u s at steady state i.e u t →...

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MPO 662 – Problem Set 1 Feel free to use symbolic computation software such as mathematica or the symbolic toolbox in matlab to carry out the algebra 1. Classify the following PDEs (a) u tt + u xx + u x = - e - kt (b) u xx - u xy + u y = 4 2. Find the characteristics of the each of the following PDEs (a) u xx + 3 u xy + u yy = 0 (b) u xx - 2 u xy + u yy = 0 3. Consider the following PDE: ∂t + U ∂x ! 2 ψ - c 2 2 ψ ∂x 2 = 0 where U and c are positive constants, and the first term is to be interpreted as: 2 ψ ∂t 2 + 2 U 2 ψ ∂x∂t + U 2 2 ψ ∂x 2 Classify this PDE Find the Characteristic lines sketch the domain of dependence and the domain of influence in the x - t plane for the point ( x 0 , t 0 ). Consider the case U > c and U < c . 4. Consider the heat equation u t = νu xx , where ν is the heat diffusion coefficient, on the interval 0 x L with the following boundary conditions u (0 , t ) = 0 and u ( L, t ) = U . Non-dimensionalize this equation using L as a length scale, and U as a u -scale, and an appropriately scaled time.

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Unformatted text preview: Find the solution u s at steady state, i.e. u t → 0. • Verify that u n = e-n 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 non-dimensional 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 non-dimensional 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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