On the Approximation of the First Eigenvalue of Some Boundary Value Problems

Pub Date : 2024-07-18 DOI:10.1134/s0965542524700465

M. Yu. Vatolkin

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Abstract

A two-point \(\left( {n - 1,1} \right)\)-type boundary value problem is investigated for the representation of eigenfunctions in the form of scalar series under the assumption that there is a functional \(\tilde {\ell }\), concentrated at one point, such that the first \(n - 1\) original boundary conditions and \(\tilde {\ell }x = 1\) turn into Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue \({{\lambda }_{ * }},\) is presented by an expansion in powers of \({{\lambda }_{ * }}.\) The equation \(\Phi (\lambda ) = 0,\) where \(\Phi (\lambda )\) is the sum of the power series in \(\lambda ,\) for finding the eigenvalues of the original problem is considered. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates for the coefficients of such power series are obtained. A function of two variables \(t\) and \(\lambda \) is determined, and a partial differential equation with conditions for this function are obtained. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.

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论某些边值问题的第一特征值的近似值

Abstract A two-point \(\left( {n - 1,1} \right)\)-type边界值问题研究了特征函数在标量级数形式下的表示，假设有一个函数 \(\tilde {\ell }\), 集中在一点上，使得第一个 \(n - 1\) 原始边界条件和 \(\tilde {\ell }x = 1\) 在这一点上变成 Cauchy 条件。与特征值 \({{\lambda }_{ * }},\) 相对应的边界值问题的特征函数是通过 \({{\lambda }_{ * }} 的幂级数展开得到的。\方程 \(\Phi (\lambda ) = 0,\)，其中 \(\Phi (\lambda )\)是 \(\lambda ,\)中的幂级数之和，用于寻找原始问题的特征值。给出了计算一些边界值问题的第一个特征值的例子。得到了对此类幂级数系数的各种估计。确定了两个变量 \(t\) 和 \(\lambda \) 的函数，并得到了带有该函数条件的偏微分方程。该函数 "截面 "的零点与原始边界值问题的特征值重合，可用于近似计算。

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