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Unformatted text preview: Quiz 4. Consider the problem u t = x k ( x ) u x ( x ) u, < x < L, t > u (0 ,t ) = 0 , u ( L,t ) = 0 , t > u ( x, 0) = f ( x ) , < x < L. (1) We assume that k ( x ) > 0. 1. Using the method of separation of variables u ( x,t ) = G ( t ) ( x ), find the equations that G and must satisfy, including their boundary conditions. 2. We assume that the corresponding eigenvalues and eigenfunctions ( n , n ( x )) are known and that the family { n ( x ) } forms a complete family of functions mutually orthogonal with respect to the scalar product ( f,g ) = R L f ( x ) g ( x ) dx . We recall the Rayleigh quotient formula = Z L 2 dx 1 Z L [ k ( ) 2 + 2 ] dx. Show that > 0 when 0 (you have to exclude = 0). 3. Solve the initial value problem. 4. What is (in general) the largest contribution in u ( x,t ) as t ? Solutions. 1. Upon plugging u ( x,t ) = G ( t ) ( x ) into (1) we obtain G ( t ) + G ( t ) = 0 , ( k ( x...
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 Fall '11
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