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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 nx L = a 2 + X n =1 a n cos nx 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 nx L dx = Z 1 cos nx 2 dx = 2 n sin nx 2 fl fl fl fl x =1 x =0 = 2 n sin n 2 . S ( x ) = a 2 + X n =1 a n cos nx 2 = 1 2 + 2 X n =1 sin n 2 n cos nx 2 . (b) [5 pts] Plot the graph of the extended f ( x ) and S ( x ) on [- 6 , 6], separately. MATH-2400 | Introduction to Differential Equations | Spring 2004 | Jeong-Rock 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 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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