On Interpolational Approximation of Nonlinear Differential Operators of the Second Order in Partial Derivatives

Speaker

Leonid Yanovich (Institute of Mathematics National Academy of Sciences of Belarus)

Description

\documentclass{article} \addtolength{\textheight}{-10mm} \addtolength{\topmargin}{-5mm} \pagestyle{plain} \def\title#1{\begin{center}#1\end{center}} \def\author#1{\centerline{\small{\textsc{#1}}}} \def\address#1#2{\begin{center}\small\emph{#1}\\E-mail: \texttt{#2}\end{center}} \def\refname{{\small References}} \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}

Primary author

Leonid Yanovich (Institute of Mathematics National Academy of Sciences of Belarus)

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