Chem Differential Eq HW Solutions Fall 2011 118

Chem Differential Eq HW Solutions Fall 2011 118 - 118...

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118 Chapter 7 The Fourier Transform and its Applications As t increases, the expression erf ± x +1 2 1 - e - t ² - erf ± x - 1 2 1 - e - t ² approaches very quickly erf ( x +1 2 ) - erf ( x - 1 2 ) , which tells us that the temperature approaches the limiting distribution 50 ³ erf ´ x +1 2 µ - erf ´ x - 1 2 µ¶ . You can verify this assertion using graphs. 17. (a) If f ( x )= · T 0 if a<x<b, 0 otherwise , then u ( x, t T 0 2 c πt ¸ b a e - ( x - s ) 2 4 c 2 t ds. (b) Let z = x - s 2 c t , dz = - ds 2 c t . Then u ( x, t T 0 2 c 2 c t ¸ x - a 2 c t x - b 2 c t e - z 2 dz = T 0 2 ³ erf ´ x - a 2 c t µ - erf ´ x - b 2 c t µ¶ . 25. Let u 2 ( x, t ) denote the solution of the heat problem with initial temperature distribution f ( x e - ( x - 1) 2 . Let u ( x, t ) denote the solution of the problem with initial distribution e - x 2 . Then, by Exercise 23, u 2 ( x, t
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