The boundary and initial conditions are t 1 t 0 t 2 t

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is a constant. The boundary and initial conditions are T (1 , t ) = 0 , T (2 , t ) = 0 , T ( x, 0) = f ( x ) (a) Show that a separation of variables solution for this equation of the form T ( x, t ) = F ( x ) G ( t ) leads to the ODEs x 2 d 2 F dx 2 + 2 x dF dx + λF = 0 , dG dt + A 2 λ G = 0
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where λ is a separation constant. (b) Show that the the ODE for F ( x ) can be written as a regular Sturm-Liouville problem, with eigenvalues λ n and eigenfunctions β n ( x ) given by 4 λ n - 1 - 2 n π ln 2 2 = 0 , β n ( x ) = 1 x sin n π ln x ln 2 , n = 1 , 2 , 3 ..... [Method: Try a solution of the form F = x σ .] (c) Obtain a series solution for the given boundary and initial conditions. 3.- (40 pts) A radioactive bar is in the form of an effectively infinitely long solid cylinder of radius r = a which is initially at temperature u ( r, t = 0) = f ( r ), where r is a radial co-ordinate in a plane normal to the bar axis. At time t = 0, the surface temperature is changed to u ( r = a, t ) = 0 at which it remains. There is subsequent heat transfer in the plane normal to the axis of the bar. The heat transfer in the radial direction inside the bar can be described by the radial heat equation ∂u ∂t - κ 2 u ∂r 2
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