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Chebyshevskii Sbornik, 2022, Volume 23, Issue 5, Pages 117–129
DOI: https://doi.org/10.22405/2226-8383-2022-23-5-117-129
(Mi cheb1259)
 

The Ritz method for solving partial differential equations using number-theoretic grids

A. V. Rodionov

Tula State Lev Tolstoy Pedagogical University (Tula)
References:
Abstract: Consider the problem
\begin{gather*} L u(\vec x) = f(\vec x), \\ u(\vec x)\big|_{\partial {G_s}}\big.=g(\vec x), \end{gather*}
where $f(\vec x), g(\vec x) \in E_s^{\alpha}$, $L$ is a linear differential operator with constant coefficients, $G_s$ is the unit cube $[0; 1]^s$.
Its solution is reduced to finding the minimum of the functional
\begin{equation*} v(u(\vec x)) =\underset{G_s}{\int\ldots\int} F\left(\vec x, u, u_{x_1}, \ldots, u_{x_s}\right) dx_1\ ldots dx_s \end{equation*}
under given boundary conditions.
The values of the functional $v(u(\vec x))$ in the Ritz method are considered not on the set of all admissible functions $u(\vec x)$, but on linear combinations
$$ u(\vec x) = W_0(\vec x) + \sum_{k=1}^{n}w_kW_k(\vec x), $$
where $W_k(\vec x)$ are some basic functions that we will find using number-theoretic interpolation, and $W_0(\vec x)$ is a function that satisfies the given boundary conditions, and the rest $W_k( \vec x)$ satisfy homogeneous boundary conditions.
On these polynomials, this functional turns into a function $\varphi (\vec w)$ of the coefficients $w_1, \ldots, w_n$. These coefficients are chosen so that the function $\varphi (\vec w)$ reaches an extremum. Under some restrictions on the functional $v(u(\vec x))$ and the basis functions $W_k(\vec x)$, we obtain an approximate solution of the boundary value problem.
Keywords: number-theoretic method, partial differential equations, variational methods.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 073-03-2022-117/7
Received: 24.07.2022
Accepted: 22.12.2022
Document Type: Article
UDC: 517
Language: Russian
Citation: A. V. Rodionov, “The Ritz method for solving partial differential equations using number-theoretic grids”, Chebyshevskii Sb., 23:5 (2022), 117–129
Citation in format AMSBIB
\Bibitem{Rod22}
\by A.~V.~Rodionov
\paper The Ritz method for solving partial differential equations using number-theoretic grids
\jour Chebyshevskii Sb.
\yr 2022
\vol 23
\issue 5
\pages 117--129
\mathnet{http://mi.mathnet.ru/cheb1259}
\crossref{https://doi.org/10.22405/2226-8383-2022-23-5-117-129}
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