testpract2——ans

testpract2——ans - Curious...

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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 ) Answer: - 2 / 3( φ ( y - x ))+5 / 3( φ (( x +2 y ) / 5))+5 / 3 Z y - x 0 ψ ( τ ) - Z ( x +2 y ) / 5 0 ψ ( τ ) Problem 2 Solve by any method: u yy - 4 u xx = sin( x - y ) , u ( x, 0) = u y ( x, 0) = 0 Answer: u ( x,y ) = 1 / 4 sin (2 y - x ) - 1 / 12 sin (2 y + x ) + 1 / 3 sin ( x - y ) 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 with the base [0 ,l ] . Similar to one solved in class (with coe cient 1 instead of 4) Problem 5 (challenging problem) 1
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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) = φ ( x ) , u t ( x, 0) = ψ ( x ) where ψ ( x ) and φ ( x ) have compact support.
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Unformatted text preview: Curious how to solve ask me or my TAs. 2 Finite Domain Problems: Problem 6 Solve: u tt-u xx + u = 0 , u (0 ,t ) = u ( π,t ) = 0 , < x < π Answer: u ( x,t ) = + ∞ X n =1 ( a n cos( p ( n 2 + 1) t ) + b n sin( p ( n 2 + 1) t )) sin ( nx ) See for two more solutions in the fourier.pdf Problem 7 Solve: u tt-4 u xx = 0 , u x (0 ,t ) = u x ( π,t ) = 0 , < x < π, u ( x, 0) = 0 , u t ( x, 0) = 1 Check your answer by direct substitution. Problem 8 Solve: u t = u xx , u x (0 ,t ) = u x (2 π,t ) = 0 , < x < 2 π, u ( x, 0) = cos( x ) Check your answer by direct substitution. Problem 9 Solve: u t = 4 u xx , u (0 ,t ) = u (2 π,t ) = 0 , < x < 2 π, u ( x, 0) = e x Check your answer by direct substitution. 3...
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testpract2&acirc;€”&acirc;€”ans - Curious...

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