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\textbf{Difference Methods for the Darboux Problem
\\
to Functional Partial Differential Equations}
\\
\textsc{Tomasz Cz{\l}api\'nski}
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We consider the following Darboux problem
\begin{gather*}
D_{xy}z(x,y)=f\left(x,y,z_{(x,y)},(D_xz)_{(x,y)},(D_yz)_{(x,y)}\right)\quad
\mbox{on}\ [0,a]\times[0,b],
\tag{1}\\
z(x,y)=\phi(x,y)\quad{\rm on}\ E^0,
\tag{2}
\end{gather*}
where $f:[0,a]\times[0,b]\times C(B,\mathbb{R})^3\to\mathbb{R}$,
$\phi\in C^1(E^0,\mathbb{R})$ and
$E^0=[-a_0,a]\times[-b_0,b]\setminus(0,a]\times(0,b]$,
$B=[-a_0,0]\times[-b_0,0]$. The operator
$[0,a]\times[0,b]\ni(x,y)\mapsto\omega_{(x,y)}\in C(B,\mathbb{R})$
defined by
$\omega_{(x,y)}(t,s)=\omega(t+x,s+y)$, $(t,s)\in B$.
represents functional dependence on unknown function and its derivatives.

We construct a wide class of difference schemes for problem (1),(2) under
the assumption that $f$ is Lipschitzean with respect to the last two
variables.
We get two convergence theorems for implicit and explicit schemes. In case
of implicite schemes we also assume that $f$ is Lipschitzean with respect
to the third variable while for explicit schemes we only assume that $f$
satisfy a nonlinear estimate of the Perron type.

The difference method that we use is generated by the method of proving the
existence of solutions of the Darboux problem in which (1),(2) is
transformed into a system of three integral functional equations.
The existence of solutions of implicite functional difference systems is
proved by means of a comparative method.


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