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Hello, I have the following problem:

Having a non-linear PDE of second order, there are the initial conditions

$F(x,0)=g(x)$,

$\partial_yF(x,0)=h(x)$.

The question is: Can the second derivatives $\partial^2_{xy}F(x,0)$ and $\partial^2_{xx}F(x,0)$ be calculated by taking derivatives on the initial conditions directly?

Thanks beforehand.

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By DEFINITION, $G_x(x,0)$ is the derivative with respect to $x$ of the function of one variable $G(x,0)$. So your second derivative is $g'$. If your $\partial_{xy}$ means $\partial_x(\partial_y)$ then your first derivative is $h'$. What does this have to do with non-linear partial differential equation, escapes me. –  Alexandre Eremenko Nov 29 '12 at 18:34
    
Thanks for the answer, I have a non-linear PDE, like $K[F(x,y),F_{x}(x,y),F_{y}(x,y),F_{xx}(x,y),F_{xy}(x,y),F_{yy}(x,y)]=0$, and the previously mentioned initial conditions. I would like to use the equation and the initial conditions to obtain $F_{yy}(x,0)$ (then the PDE would be an algebraic equation). The ingredients missing are $F_{xx}(x,0),F_{xy}(x,0)$, so I wonder if they can be obtained from the initial conditions. –  crismalo Nov 29 '12 at 21:49

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