430F09exam1sol - Math 430/530 Midterm Exam#1 Fall 2009...

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Math 430/530 Midterm Exam #1 Fall 2009 Solutions Instructions: Everyone should do the first three problems. Problem 4 is only for Math 430 students. Problem 5 is only for Math 530 students. Each question is worth a total of ten points. Partial credit is possible. The last page of this exam contains formulas and other information you may find useful! 1) When the temperature u between two concentric spheres is spherically symmetric ( u = u ( r, t )) and thermal properties are constant, u satisfies the following PDE: ∂u ∂t = k r 2 ∂r r 2 ∂u ∂r Suppose the radii of the concentric spheres bounding a solid region are 2 and 4. Suppose it is known that the steady-state temperature is spherically symmetric and satisfies u (2) = 10 and ( ∂u ) / ( ∂r )(2) = 3 (in appropriate units). Showing your work, find the steady-state temperature on the sphere of radius 4. Solution: The steady-state temperature u is independent of t , so the PDE above becomes the following ODE: 0 = k r 2 d dr r 2 du dr Canceling k/r 2 , this simplifies to 0 = d dr r 2 du dr This implies that r 2 du dr = C 1 or du dr = C 1 r 2 Integrating both sides, we get that u ( r ) = - C 1 r + C 2 where C 1 and C 2 are constants. Using the given information that u 0 ( r ) = 3, we obtain that C 1 = 12. Then substituting this and the other piece of
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given information ( u (2) = 10) into the formula for u , we get that C 2 = 16 . Finally evaluating u ( r ) = - 12 /r + 16 at r = 4, we get that the steady-state temperature on the outer sphere is 13. 2) Write a formula for the solution of the following one-dimensional heat equation problem: ∂u ∂t = 3 2 u ∂x 2 , 0 x 4 u (0 , t ) = u (4 , t ) = 0 , for all t > 0 u ( x, 0) = 2 , 0 x 4 Recall that when variables are separated by u ( x, t ) = g ( t ) φ ( x ) for the general one-dimensional (homogeneous, constant thermal properties) heat equation u t = ku xx , we get that g 0 = - λkg , and φ must satisfy an eigenvalue problem, with boundary conditions coming from the boundary conditions of the PDE problem. Recall also that the eigenvalues for the following problem d 2 φ dx 2 = - λφ φ (0) = φ ( L ) = 0 are λ = ( nπ/L ) 2 for positive integers n with corresponding eigenfunctions sin( λx ). Solution: Using our usual notation, k = 3 and L = 4 in this problem. Using the given information, the eigenvalues for the spatially-dependant part of the problem are λ = ( nπ/ 4) 2 where n is any positive integer and the corresponding eigenfunctions are sin( nπx/ 4). For each eigenvalue λ , the solution of the time-dependent problem is e - 3( nπ/ 4) 2 t . So the solution to the problem is an infinite series of the form u ( x, t ) = X n =1 A n e - 3( nπ/ 4) 2 t sin nπx 4 · It remains to find the coefficients A n
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