Differential Equations Lecture Work Solutions 227

Differential Equations Lecture Work Solutions 227 - 2a. uxx...

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2a. u xx u yy +3 u x 2 u y + u =0 U = ue ( αξ + βη ) A =1 B =0 C = 1 4 = 4 hyperbolic dy dx = ± 2 2 = ± 1 ξ = y x η = y + x u x = u ξ + u η u y = u ξ + u η u xx = ( u ξξ + u ξη )+( u ξη + u ηη )= u ξξ 2 u ξη + u ηη u yy = u ξξ +2 u ξη + u ηη 4 u ξη 3 u ξ +3 u η 2 u ξ 2 u η + u =0 4 u ξη 5 u ξ + u η + u =0 U = ue ( αξ + βη ) u = Ue ( αξ + βη ) u ξ = U ξ e ( αξ + βη ) + αUe ( αξ + βη ) u η = U η e ( αξ + βη ) + βUe ( αξ + βη ) u ξη = U ξη e ( αξ + βη ) + βU ξ e ( αξ + βη ) + αU η e ( αξ + βη ) + αβUe ( αξ + βη ) 4 U ξη 4 βU ξ 4 αU η 4 αβU 5 U ξ 5 αU
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