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Practice exam solutions 3

Practice exam solutions 3 - Math 335 Practice Exam 3 Key...

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Math 335 Practice Exam 3 Key 1.) [10 points] A metal bar of length π initially has temperature u ( x ,0) = − x at position x, where x=0 is the left end of the bar. At time t=0, the two ends of the bar are wrapped in ice cubes with constant temperature 0 degrees Celsius. Find the temperature u(x,t) of the bar assuming a thermal diffusivity constant k=3.

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1 2.) [10 points] Find the Fourier series of gG±² = |±| on the interval G−³,³² . What value does the Fourier series converge to at ± = ³ ?
2 3.) [10 points] State in words the physical situation that the equations below describe: g GG = 4g ±± , g²0,³´ = 0 , g²10,³´ = 0 , g²µ, 0´ = 20 , ¶· ¶G ²µ, 0´ = 0 . Then solve the boundary value problem. A vibrating string with tension constant a=2 ( ¸ ¹ = 4´ has length 10 and the ends are clamped at the x-axis. The string is held at a position 20 above equilibrium and released from rest (zero velocity).

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3 4.) [10 points] The ends of a metal bar of length L with thermal diffusivity constant k are insulated so that no heat flows in or out through the ends.
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Practice exam solutions 3 - Math 335 Practice Exam 3 Key...

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