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The Ritz method for solving partial differential equations using number-theoretic grids
A. V. Rodionov Tula State Lev Tolstoy Pedagogical
University (Tula)
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.
Received: 24.07.2022 Accepted: 22.12.2022
Citation:
A. V. Rodionov, “The Ritz method for solving partial differential equations using number-theoretic grids”, Chebyshevskii Sb., 23:5 (2022), 117–129
Linking options:
https://www.mathnet.ru/eng/cheb1259 https://www.mathnet.ru/eng/cheb/v23/i5/p117
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Abstract page: | 79 | Full-text PDF : | 65 | References: | 20 |
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