Solution of Boundary-Value Problems using Kantorovich Method

¹ Joint Institute for Nuclear Research, Dubna, Russia
² Belgorod State University, Belgorod, Russia
³ National University of Mongolia, UlaanBaatar, Mongolia
⁴ Saratov State University, Saratov, Russia

^a e-mail: gooseff@jinr.ru
^b e-mail: vinitsky@theor.jinr.ru

Published online: 9 February 2016

Abstract

We propose a computational scheme for solving the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated as solutions of the parametric eigenvalue problem for an ordinary second-order differential equation. As a result, the initial problem is reduced to a boundary-value problem for a set of self-adjoint second-order differential equations for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method with Hermite interpolation polynomials. The effciency of the calculation scheme is shown by benchmark calculations for a square membrane with a degenerate spectrum.