Existence And Uniqueness for a Third Order Non Linear Partial Differential Equat

Cover Existence And Uniqueness for a Third Order Non Linear Partial Differential Equat
Existence And Uniqueness for a Third Order Non Linear Partial Differential Equat
Chester B Sensenig
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Since w(x, y, t) co = M 1 ■ - 1 - — k' (\>/z +y 2 )d£, we have w = and hence it J /g ? X -co yz +y w - X w = 0. Therefore w = c, e~ ' + c e y . Since w is yy 12 bounded, then c ? = 0. Since w(x, 0, t) =1, c, = 1 and w = e" y .
Replacing h(£/t, t )+X 2 (a£+b) by 1 in (II. 1) and (II. 5) we see that v = and v (x, y, t) = w(x, y, t) = e y . Hence v = — k e" * + v-. Y + Vp. Since v is bounded, then v, = 0. Since v(x, 0, t) = 0, then v ? = -| and v = i (e~ Ay -l).
X X p We now have A v - X v 88 "Si
...U Ag( x »y;^, / ?)[h(?, '?, t)-h(x, y, t)+aX 2 (C-x)] d^dl + w(x, y, t)[h(x, y, t)+X 2 (ax+b)] g(x, y;^, ^)[h(5, / ?, t)+X^(a^+b)]d?d^. Using the fact that -sj ^>0 ■ . .
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: - - '■ - - .
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- 38 As( x >y;£/y;ZA) v © have /\v- \ v = [h(x, y, t)+X 2 (ax+b)] [w(x, y, t )-\ 2 v(x, y, t ) ] = h(x, y, t )+X 2 (ax+b).
Since u = v-w+ax+b, then u has continuous second derivatives with respect to x and y at (x_, y . T ), and Au(x_, y_, t ) - o o o ■ o o o x2u(x o' y o' t o ) = h ( x 'y o ' t o )+x2(ax o +b) "°" x2(ax o +b) = h(x o' y o' t o ) * This completes the proof of Theorem II.


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