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.
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.
| № | Author name | position | Name of organisation |
|---|---|---|---|
| 1 | Abdukarimov A.. | teacher | TSTU |
| 2 | Khaldybaeva I.. | teacher | TSTU |
| 3 | Kuralov B.. | teacher | TSTU |
| 4 | Askarova A.. | functionary | Digital technologies and artificial intelligence research Institute |
| № | Name of reference |
|---|---|
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| 3 | A. Abdukarimov. A new proof of the addition theorem for integral-differential equations of dynamic problems of hereditarily deformable systems. “Problemy mechanical”, 2004. 19. |
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| 9 | A. Abdukarimov., F.B. Badalov., S. Babazhanova. Numerical-analytical solution of dynamic problems of thin-walled and bar structures made of composite materials. “Computational experiment, mathematical modeling and their application in applied mathematics and mechanics”, 1994. 9 |
| 10 | Agavai R.P., Lakshmikantham V., Nietto J.J. On the concept of solution for fractional differential equations with uncertainty // Nonlinear Analysis. 2010. 72 |
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| 12 | A. Abdukarimov, Zhunisbekov A. Khodjibergenov D. Research of random oscillations of hereditarily deformable systems. “Proceedings of the international scientific and practical conference dedicated to the 70th anniversary of the South Kazakhstan State University. M. Auezov”. Shymkent-2013. 36 |
| 13 | I.K. Pokhodnya., V.Z. Turkevich. Physical and technical problems of modern materials science. “Kiev-Academic Periodics”, 2013. 39. |
| 14 | Bazhanov V.L. Mechanics of a deformable solid body: a textbook for universities / Moscow: Yurait Publishing House, 2020. 178 |
| 15 | A. Abdukarimov., I. Khaldybaeva., E.N. Gribanov. Solutions of the problem of random vibrations in hereditary-deformable systems using impulsive functions. “Technical science and innovation”, 2020. 156 |
| 16 | S.N. Askharov. Boundary value problems for singular integro-differential equations with monotone nonlinearity. “Materials of the International Conference Voronezh Winter Mathematical School”. 2021. 42 |
| 17 | M.T. Jenaliev., M.I. Ramazanov., N.K. Gulmanov. The solution of special Volterra integral equation of the second kind. “Materials of the International Conference Voronezh Winter Mathematical School”. 2021. 109 |
| 18 | A. Abdukarimov., I. Khaldybaeva., B.A. Kuralov. Forced oscillations of rectangular plates from dissipative nonhomogeneous composite materials. “Technical science and innovation”, 2022. 184 |