Stability of mathematical models of the main problems of the anisotropic theory of elasticity

A. V. Yudenkov   ; A. M. Volodchenkov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 1, pp. 112-124

Résumé

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the H\"older space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

Détail

Keywords: boundary value problem, analytic function, elasticity theory, Fredholm equation.

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