testpract2 - with the base [0 ,l ] . Problem 5 (challenging...

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In nite Domain Problems ( -∞ < x,y < + ): Problem 0 Find the region in xy plane where the equation (1 - x ) u xx + 2 y u xy + (1 + x ) u yy = 0 is elliptic, hyperbolic, parabolic. Problem 1 Solve by any method: u yy - u yx - 2 u xx = 0 , u ( x, 2 x ) = φ ( x ) , u y ( x, 2 x ) = ψ ( x ) Problem 2 Solve by any method: u yy - 4 u xx = sin( x - y ) , u ( x, 0) = u y ( x, 0) = 0 Problem 3 Solve using Green's formula: u yy - u xx = cos( x + y ) , u ( x, 0) = cos( x ) , u y ( x, 0) = sin( x ) Check your answer by direct substitution. Problem 4 Use the energy method to prove that if u xx - 4 u yy = 0 , u ( x, 0) = u y ( x, 0) = 0 , 0 x l then u ( x,y ) 0 inside the triangle domain of dependence
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Unformatted text preview: with the base [0 ,l ] . Problem 5 (challenging problem) Use the energy method to prove that for t big enough 1 2 Z + - ( u t ) 2 dx = 1 2 Z + - ( u x ) 2 dx (equipartition of kinetic and potential energies for large times) if u xx-u tt = 0 , u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) where ( x ) and ( x ) have compact support. 1...
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