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Unformatted text preview: Exam 2  March 26, 2004  Solution 1 1. A function f ( x ) is originally defined on [ 2 , 2] by f ( x ) = , 2 ≤ x < 1 , 1 , 1 ≤ x ≤ 1 , , 1 < x ≤ 2 , and extend it to the whole real line to have the period T = 4. (a) [10 pts] Calculate the Fourier series S ( x ) of f ( x ). [Hint. Utilize the evenness or oddness of the function f .] Since T = 4 = 2 L , we have L = 2. Since f ( x ) is an even function, the Fourier series S ( x ) of f ( x ) is a cosine series: f ( x )“ = ” S ( x ) = a 2 + ∞ X n =1 a n cos nπx L = a 2 + ∞ X n =1 a n cos nπx 2 , where a = 2 L Z L f ( x ) dx = Z 2 f ( x ) dx = Z 1 1 dx = 1 , a n = 2 L Z L f ( x )cos nπx L dx = Z 1 cos nπx 2 dx = 2 nπ sin nπx 2 fl fl fl fl x =1 x =0 = 2 nπ sin nπ 2 . ∴ S ( x ) = a 2 + ∞ X n =1 a n cos nπx 2 = 1 2 + 2 π ∞ X n =1 sin nπ 2 n cos nπx 2 . (b) [5 pts] Plot the graph of the extended f ( x ) and S ( x ) on [ 6 , 6], separately. MATH2400  Introduction to Differential Equations  Spring 2004  JeongRock Yoon Exam 2  March 26, 2004  Solution 2 2. Let a 20 centimeter long metal rod whose thermal diffusivity is 2 cm 2 /sec is given. Suppose that the left end is kept as 0 ◦ Celsius all the time, the right end is kept as 10 ◦ Celsius all the time and the lateral surface is insulated. Assume that the initial temperature in the rod is given by a piecewise linear function consistent with the preceding boundary conditions and being 20 ◦ Celsius at the center point of the rod. (a) [6 pts] Establish the differential equation model for the temperature u ( x,t ) in the rod at any time t ≥ 0 with appropriate boundary conditions and initial condition. Since the thermal diffusivity is α 2 = 2 and the length of rod is L = 20, we have u t = 2 u xx , < x < 20 , t > , heat equation , u ( x, 0) = f ( x ) , < x < 20 , 1 initial condition , u (0 ,t ) = 0 , u (20 ,t ) = 10 , t > , 2 boundary conditions...
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 Spring '04
 Yoon
 Differential Equations, Equations, Fourier Series, Boundary value problem, Partial differential equation

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