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A NEW PROOF OF THE ADDITION THEOREM FOR INTEGRO-DIFFERENTIAL EQUATIONS OF DYNAMIC PROBLEMS OF HEREDITARILY DEFORMABLE SYSTEMS

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Technical science and innovation

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ARTICLE ANNOTATION

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F. Tricomi [9] first proposed the addition theorem for second-order ordinary differential equations. The method proposed here is a natural generalization of this method for IDE functions of the sine and cosine of a fractional order, since in the special case for R = 0 the well-known addition theorem proposed by F. Tricomi follows [9]. An exact solution of an integrodifferential equation (IDE) with initial conditions and arbitrary Abel-type hereditary kernels can be constructed by the method of fundamental systems of solutions. Using the exact solution without proving the addition theorem to study real oscillatory and wave processes for t > 1 leads to certain computational difficulties. The exact solution of the integro-differential equation (IDE) will make it possible to detect a number of new mechanical effects, in particular, vibrations, displacements and deformations of any mechanical systems, such as shell structures, under the action of a constant external load occur near the creep function curve, and the stress near the relaxation function and decay over time along this curve. These results serve as a test for checking the accuracy of solutions of numerical and approximate analytical methods for solving IDEs of dynamic problems in the theory of viscoelasticity. In this paper we present a new simpler proof of this theorem.

AUTHORS

A.Abdukarimov

TSTU

I.Khaldybaeva

TSTU

B.Kuralov

TSTU

A.Askarova

Digital technologies and artificial intelligence research Institute

Tags

# mathematical model# anisotropic materials# integro-differential equation (I# relaxation kernel# forced vibrations# bed coefficient# reactive pressure

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Article Identifiers

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https://eroi.uz/11.0008/A0423-0030

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