homework3 - 1.6.1(b What is the type of the following...

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Problem 1 (not from the textbook). Solve the initial value problem: u x + 2 xu y = 2 xu, u (0 ,y ) = y 2 by two di erent methods (using parametrization of the characteristic curve such as γ ( s ) = ( x ( s ) ,y ( s )) and such as γ ( x ) = ( x,y ( x )) . Hint - see pdf notes for the second week lectures ) Sketch the initial value line and some characteristics. Find the domain of well-posedness of the problem.
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Unformatted text preview: 1.6.1 (b) What is the type of the following equation ? 9 u xx + 6 u xy + u yy + u x = 0 1.6.2 Find the regions in the xy plane where the equation (1 + x ) u xx + 2 xy u xy-y 2 u yy = 0 is elliptic, hyperbolic, or parabolic. Sketch them. 2.1.10 Solve u xx + u xt-20 u tt = 0 , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) . 1...
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