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\begin{document}
\title{\textsc{On Interpolational Approximation
of Nonlinear Differential Operators of the Second Order in
Partial Derivatives{}}\footnote{This work is supported by
Belarusian Republican Foundation for Fundamental Research (project
16D-002).}}
\author{L.~A.~Yanovich and M.~V.~Ignatenko}
\address{Institute of Mathematics, \\ National Academy of Sciences of Belarus, \\
Surganova Str. 11, \\
220072 Minsk, Belarus}{yanovich@im.bas-net.by, ignatenkomv@bsu.by}
\bigskip
\small
We consider the differential operators $F:C^{2} \left(T\times
S\right)\to Y$ of the second order in partial derivatives of the
form $$ F\left(x\right)=f\left(t,s,x\left(t,s\right),x'_{t}
\left(t,s\right),x'_{s} \left(t,s\right),x''_{t^{2} }
\left(t,s\right),x''_{t,s} \left(t,s\right),x''_{s^{2} }
\left(t,s\right)\right), \eqno (1)$$ where $x'_{t}
\left(t,s\right)=\frac{\partial x\left(t,s\right)}{\partial t
},$
$x''_{t^{2} } \left(t,s\right) =\frac{\partial ^{2}
x\left(t,s\right)}{\partial t^{2} },$ $x''_{t,s} \left(t,s\right)=
\frac{\partial ^{2} x\left(t,s\right)}{\partial t \partial s },$
the derivative $x''_{t,s}
\left(t,s\right)=x''_{s,t} \left(t,s\right)$, the space $C^{2}
\left(T\times S\right)$ is the space of two times continuously
differentiable on $T\times S\subseteq R^{2} $ functions
$x\left(t,s\right),$ the function $y=f\left(t,s,u_{0} ,u_{1}
,...,u_{5} \right)$ is defined on a rectangle $\Omega =T\times
S\times T_{0} \times T_{1} \times \cdots \times T_{5} ,$ $T_{i} $
are sets of the number line $\left(i=0,1,...,5\right)$, and $Y$ is
a function space.
Here is the Lagrange interpolation formula for the operators (1):
$$L_{n} \left(F;x\right)=F\left(x_{0} \right)+ \sum _{k=1}^{n} \,
\int _{0}^{1} \sum _{i,j=0; \, i+j\le 2} ^{2} \frac{\partial
}{\partial \left(\frac{\partial ^{i+j} \upsilon _{k} }{\partial
t^{i}
\partial s^{j} } \right)} F\left(\upsilon _{k} \left(t,s,\tau
\right)\right) \times $$ $$\times\frac{\partial ^{i+j} }{\partial
t^{i}
\partial s^{j} } \left\{\frac{l_{n,k}
\left(x\left(t,s\right)\right)}{\sigma _{n}
\left(x\left(t,s\right)\right)} \left(x_{k} \left(t,s\right)-x_{0}
\left(t,s\right)\right)\right\}d\tau , \eqno (2)$$ where the
functions $\upsilon _{k} =\upsilon _{k} \left(t,s,\tau
\right)=x_{0} \left(t,s\right)+\tau \left(x_{k}
\left(t,s\right)-x_{0} \left(t,s\right)\right)$
$\left(k=1,2,...,n\right),$ $l_{n,k} \left(x\right)$ are
fundamental polynomials of the $n$-degree with respect to the
Chebyshev system of functions $\mathop{\left\{\varphi _{k}
\left(x\right)\right\}}\nolimits_{k=0}^{n} $, $l_{n,k} (x_{j}
)=\delta _{kj} $ is the Kronecker symbol $(k,j=0,1,...,n),$ and
$\sigma _{n} \left(x\right)=\sum _{k=0}^{n}l_{n,k} \left(x\right)
\, $ is a constant or a variable value. The polynomial (2)
satisfies to the following interpolation conditions:
\[L_{n} \left(F;x_{k} \right)=F\left(x_{k} \right), \left(k=0,1,...,n\right).\]
For the interpolation error $r_{n}
\left(x\right)=F\left(x\right)-L_{n} \left(F;x\right)$, where
$L_{n} \left(F;x\right)$ is interpolation polynomial (2), the
following representation holds:
\[r_{n} \left(x\right)=\sum _{k=1}^{n+1} \, \int _{0}^{1}
\sum _{i,j=0; \, i+j\le 2} ^{2}\frac{\partial }{\partial
\left(\frac{\partial ^{i+j} \upsilon _{k} }{\partial t^{i} \partial s^{j} } \right)}
F\left(\upsilon _{k} \left(t,s,\tau \right)\right) \times \]
\[\times \frac{\partial ^{i+j} }{\partial t^{i} \partial s^{j} }
\left\{\left(\frac{l_{n+1,k} \left(x\left(t,s\right)\right)}{\sigma _{n+1}
\left(x\left(t,s\right)\right)} -\frac{l_{n,k} \left(x\left(t,s\right)\right)}{\sigma _{n}
\left(x\left(t,s\right)\right)} \right)\left(x_{k} \left(t,s\right)-x_{0}
\left(t,s\right)\right)\right\}d\tau ,\]
where $x_{n+1} =x,$ $l_{n,n+1} (x)\equiv 0$.
Some other interpolation formulas for the operator (1) are also
constructed.
\end{document}