H8_130 - Math 415 Homework 8 10.5.7 Solve 100 u xx = u t<...

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Unformatted text preview: Math 415 Homework 8 10.5.7 Solve 100 u xx = u t , < x < 1 , t > 0; u (0 ,t ) = 0 = u (1 ,t ) , t > 0; u ( x, 0) = sin2 πx- sin5 πx, ≤ x ≤ 1 . The general solution to this heat equation problem is u ( x,t ) = ∞ X n =1 c n e- 100 n 2 π 2 t sin nπx. Looking at the initial condition, we must have u ( x, 0) = ∞ X n =1 c n sin nπx = sin2 πx- sin5 πx. If two sums of sine functions with similar inputs are equal, then the coefficients must be equal. I.e. c 1 = 0 since there is no sin πx term on the right hand side, c 2 = 1, c 5 =- 1, and c i = 0 for i 6 = 2 or 5. So our final solution is u ( x,t ) = e- 400 π 2 t sin2 πx- e- 2500 π 2 t sin5 πx. 10.5.10 Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0 ◦ C for all t > 0. Find an expression for the temperature u ( x,t ) if the initial temperature is the function u ( x, 0) = x, for 0 ≤ x < 20 , 40- x for 20 ≤ x ≤ 40 ....
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H8_130 - Math 415 Homework 8 10.5.7 Solve 100 u xx = u t<...

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