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Unformatted text preview: Practice Final, APMA 4200, 2011. PROBLEM 1. Consider the function g ( x ) = x for x (0 , ) ,- x for x (- , 0) . 1.1. Show that the function is even. 1.2. Calculate its Fourier coefficients. We recall that a 2 periodic function can be represented as f ( x ) = a + X n =1 a n cos( nx ) + b n sin( nx ) , b n = 1 Z - f ( y )sin( ny ) dy, a = 1 2 Z - f ( y ) dy, a n = 1 Z - f ( y )cos( ny ) dy. 1.3. Is the function g ( x ) extended by periodicity a continuous function on the whole line x (- , )? 1.4. Justify that you can differentiate term by term in the Fourier series expansion of g ( x ) and deduce the Fourier series expansion of the function h ( x ) = 1 for x (0 , ) ,- 1 for x (- , 0) . PROBLEM 2. Consider the heat equation u t = k 2 u x 2 , < x < , t > u x (0 ,t ) = 0 , u x ( ,t ) = 0 , t > , u ( x, 0) = x for 0 < x < . (1) 2.1. Using the method of separation of variables u ( x,t ) = G ( t ) ( x ), write the ordinary differential equations that G and must satisfy (including boundary conditions for ). 2.2. Solve for G ( t ) knowing G (0). 2.3. Solve the eigenvalue problem for (we assume to save time that 00 + = 0 with the proper boundary conditions has no non-trivial solution when < 0). 2.4. Write down the most general solution u ( x,t ) that can be constructed from the above method of separation of variables. Justify the formula....
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This note was uploaded on 02/05/2012 for the course APMAE 4200 taught by Professor R during the Fall '11 term at Columbia.
- Fall '11