Practice-Midterm-2011

Practice-Midterm-2011 - E4200 Practice Midterm 1. Let f ( x...

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Unformatted text preview: E4200 Practice Midterm 1. Let f ( x ) be the 2- periodic function defined on (- 1 , 1) by f ( x ) = ( 1 + x on (- 1 , 0) 1- x on (0 , 1) . 1.1. Sketch the graph of f ( x ). 1.2. Show that f ( x ) is even. 1.3. Show that Z 1 (1- x )cos( nx ) dx = if n 2 and n is even 2 ( n ) 2 if n 1 and n is odd . 1.4. We recall that the 2- periodic function f ( x ) can be represented by f ( x ) = a + X n =1 a n cos( nx ) + b n sin( nx ) , where a = 1 2 Z 1- 1 f ( x ) dx, a n = Z 1- 1 f ( x )cos( nx ) dx, b n = Z 1- 1 f ( x )sin( nx ) dx. Calculate the coefficients a , a n and b n for n 1. 1.5 Calculate f ( x ) and its Fourier decomposition. 2. Let 0 < 2 c/L , where c > 0 and L > 0. We consider the problem 2 u t 2 + u t = c 2 2 u x 2 x (0 ,L ) , t > u ( t, 0) = 0 , u ( t,L ) = 0 , t > u (0 ,x ) = 0 , u t (0 ,x ) = g ( x ) , x (0 ,L ) . 2.1. Consider elementary solutions of the form u ( t,x ) = G ( t ) ( x ), where u ( t,x ) solves the above equation except for the initial conditions. Find the equations (including possible boundary conditions) that G ( t ) and ( x ) must satisfy. 2.2. For the equation 00 + = 0 on (0 ,L ), the Rayleigh quotient states that Z L 2 ( x ) dx = Z L ( ) 2 ( x ) dx + (0) (0)- ( L ) ( L ) . Show that the eigenvalues associated to the problem for obtained in 2.1 must satisfy > 0. Solve the equation for ( , ). 2.3. We recall that the solutions to y 00 + y + 2 y = 0 for < 2 are given by y = e- t/ 2 ( A cos( t ) + B sin( t )) , = 1 2 q 4 2- 2 . Recalling that 0 < < 2 c/L , solve for G ( t ) obtained in 2.1....
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Practice-Midterm-2011 - E4200 Practice Midterm 1. Let f ( x...

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