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AMath350.F12.A8.Sol

# Never occurs sin u 0 equation is parabolic if sin2 x

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Unformatted text preview: = ( 2 u⇠ ˆ @y @y 2 u⌘ ) = ˆ 2u⌘⌘ ) = ˆ 6u⇠⇠ ˆ 2 (ˆ⇠⇠ ⇠y + u⇠⌘ ⌘y ) u ˆ 8u⇠⌘ ˆ 2u⌘⌘ . ˆ 2 (ˆ⌘⇠ ⇠⌘ + u⌘⌘ ⌘y ) u ˆ AMATH 350 Page 3 Assignment #8 Solutions - Fall 2012 = 2 ( 2u⇠⇠ ˆ 2u⇠⌘ ) ˆ 2 ( 2u⌘⇠ ˆ 2u⌘⌘ ) = 4ˆ⇠⇠ + 8ˆ⇠⌘ + 4ˆ⌘⌘ . ˆ u u u We’re now ready to rewrite the given PDE, and integrate: 3 uxx + 2uxy + uyy = 0 4 =) 3 2u⌘⌘ )+ (4ˆ⇠⇠ + 8ˆ⇠⌘ + 4ˆ⌘⌘ ) = 0 ˆ u u u 4 Equation is hyperbolic if ) 2 (x) >41. Never occurs. =sin u⇠⌘ = 0 ˆ Equation is parabolic if sin2 (x) = 1 ) x = 2 + k ⇡ , k Z Z Equation is hyperbolic if2=) (x) > ⇠⌘ = 0 occurs. sin2 u1. Never ˆ Equation is elliptic if sin (x) < 1 2 Equation is parabolic if if sin2 () => 1. ) x = occurs. , k Z Z Equation is hyperbolic sin (x x) u1 = g (⇠ ) Never 2 + k ⇡ =) ˆ⇠ 22 Equation is elliptic if sin sin )(<) 1 1 ) x = + k ⇡ , k Z Equation is parabolic if (x x = Z 2 =) is elliptic ) = G2((x)+...
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