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** Description:** Differential taylorian transformations are used in this paper to create a series of symbolic models of physical processes that are appropriate to represent the Fredholm integral equation of the first kind, and demonstrate the advantages of using these transforms on three different models. The use of differential taylorian transforms reduces the solving of a complex problem to simpler, thereby reducing computational complexity, since it allows us to represent with an acceptable accuracy the complex inverse problem with a simpler model, a pronounced system of linear algebraic equations of small dimension. There was found unknown function of dispensing of electrical potential in semi-conductor with the help of differential transformation. The unknown function of dispensing is a higher bound of integration in integral equation of Fredholm of first kind. Assembled differential model of this equation that allow to find unknown function of dispensing of potential. With the help of differential taylorian transformations, error stabilized experimental data model, based on the Fredholm integral equations is also obtained, which is suitable for studying the thermophysical characteristics of the components of the vehicles' nodes, in particular the study of the thermal state of the engines; overheating of brake pads, bearing units, etc. In this case, the heat equation is a differential equation in partial parabolic derivatives. On the basis of differential transformations the symbolical model is developed for definition thermal characteristics of materials, in particular, parameter temperature conductivity steady against mistakes in initial data. The considered problem describes a high-temperature mode with ablation of weight of the fused substance, for example, the process of destruction of the protective layer of aerospace transport. The approach considered in this work can be used in other tasks, when physical processes should be considered in an integral form. The use of differential taylorian transforms allows you to reduce the complex task to a simpler, thereby reducing computational complexity.

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Keywords:
** differential taylorian transformations, the Fredholm integral equation of the first kind, incorrectness, inverse coefficient problem of temperature-conductivity

1. Zasjadko, A.A. (2002), “Differentsial'no-teylorovskaya model' zadachi vosstanovleniya v spektroskopiji” [Differential-Taylor model of the recognition problem in spectroscopy], Electronic modeling, No. 6(20), pp. 97-105.

2. Zasjadko, A.A. and Pochka, S.I. (2006), “Metod dyferentsialʹnykh peretvorenʹ dlya modelyuvannya protsesu vidnovlennya dvovymirnykh syhnaliv” [Differential transformation method for simulation of two-dimensional signal recovery process], Announcer of the Khmelnytsky National University. Series Technical Sciences, No. 1, pp. 214-219.

3. Ivasishin, S.D. (1987), “Lineynyye parabolicheskiye granichnyye zadachi” [Linear parabolic boundary value problems], High School, Kyiv, 73 p.

4. Korolyuk, S.L., Korolyuk, S.S., Tsarenko, I.M., Tarko, O.L. and Galochkin, A.V. (2001), “Sobstvennyye poluprovo-dniki gruppy kak perspektivnyye materialy dlya radiatsionno stoykoy elektroniki” [Own semiconductors of the group as promising materials for radiation-resistant electronics], Technology and design in electronic equipment, No. 6, pp. 3-5.

5. Pukhov, G.Ye. (1990), “Differentsialnie spektri i modeli” [Differential spectrums and models], Scientific thought, Kyiv, 184 p.

6. Frolov, G.A. and Baranov, V.L. (2007), “Dinamika progreva tverdogo tela pri teplovom razrushenii poverkhnosti” [Dynamics of heating a solid with thermal destruction of the surface], Physical Engineering Journal, No. 6(80), pp. 30-43.

7. Lie-jun, Xie, Cai-lian, Zhou and Song, Xu (2016), An e_ective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method, Springer Plus, 5:1066, pp. 1-21.

8. Kader, A.H. Abdel, Latif, M.S. Abdel and Nour, H.M. (2016), Exact solution of fin problem with linear temperature-dependent thermal conductivity, Journal of Applied Mathematics and Computational Mechanics, No. 15(4), pp. 51-61.

9. Hatami, M., Ganji, D.D. and Sheikholesami, M. (2016), Differential Transformation Method for Mechanical Engineering Problems, Academic Press is an imprint of Elsevier, 410 p., available at: https://books.google.com.ua/books?isbn=0128053402.

10. Ghasemi, Seiyed E., Hatami, M. and Ganji, D.D. (2014), Thermal analysis of convective fin wit h temperature-dependent thermal conductivity and heat generation, Case Studies in Thermal Engineering, No. 4, pp. 1-8. http://dx.doi.org/10.1016/j.csite.2014.05.002.

11. Mosayebidorcheh, S., Ganji, D.D. and Farzinpoor, M. (2014), Approximate solution of the nonlinear heat transfer equation of a ﬁn with the power-law temperature-dependent thermal conductivity and heat transfer coefﬁcient, Propulsion and Power Research, No. 3(1), pp. 41-47. http://dx.doi.org/10.1016/j.jppr.2014.01.005.

12. Szénási, S. and Felde, I. (2017), Configuring Genetic Algorithm to Solve the Inverse Heat Conduction Problem, Acta Polytechnica Hungarica, No. 6 (14), Budapest, pp. 133-152.

Zasiadko, A.A. (2019), “Symvolichni modeli fizychnykh protsesiv, shcho opysuiutsia intehralnym rivnianniam Fredholma pershoho rodu” [Symbolic models of physical processes described the Fredholm integral equation of the first kind],