Solution assume that ux y f xgt then ut f xg0

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Unformatted text preview: der the PDE 8. Consider the PDE uxx + Dux + Euy + F u = g (x, y ) uxx + Du + Euy where D, E, F are constants andx E = 0. + F u = g (x, y ) (1) (1) where u x, F are x+ y v (x, ). Then (a) LetD,(E,y ) = e constantsyand E = 0. (a) ux =u(x,+ ) ( v + +xy , (uy y ).e Then( v + vy ), uxx = e x+ y ( 2 v + 2 vx + vxx ) Let e x y y = e x v )v x, = x+ y x ux = e x+ y v + vx ) uy gives + y ( v + vy ), uxx = e x+ y ( 2 v + 2 vx + vxx ) Substituting( into the, DE = e y e x+Substitutingxinto theDe xgives v +vx )+Ee x+ y ( v +vy )+F e x+ y v = g (x, y ) ( 2v +2 v +vxx )+ DE + y ( e x+ y ( 2v +2 terms:xx )+De x+ y ( v +vx )+Ee x+ y ( v +vy )+F e x+ y v = g (x, y ) Collecting vx +v Collecting(2 + D)vx + Evy + ( 2 + D + E + F )v = e x y g (x, y ) vxx + terms: 2 xy vxx + 2 (b) Choosing (2 =+ D)vx and y + (( + D D+ E F+/E)v = D2 /4 g ()/E) will D/2 + Ev = ) F = (e F x, y 2 second and fourth (b) eliminate the= D/2 and = (terms, giving the/E = (D2 /4 for)v (x,will Choosing D F ) following DE F /E y ): eliminate t...
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This document was uploaded on 02/09/2014.

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