By using the N th order approximation of temperature function and its first derivative by Legendre polynomials, we construct the solution of problem of heat conduction for a thin-walled shallow isotropic shell and deduce the system of resolving equations for the N th approximation. For the first and third approximations, we solve this problem for the case of concentrated heat sources. In the third approximation, we also construct the plots of the dependences of temperature on the distance to the heat source and on the curvature of the shell under the conditions of symmetric or asymmetric stationary heat exchange.
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N. S. Bondarenko and A. S. Gol’tsev, “Solution of the heat-conduction problem for anisotropic plates under concentrated thermal loading using Legendre polynomials,” Mat. Metody Fiz.-Mekh. Polya, 52, No. 4, 216–226 (2009); English translation: J. Math. Sci., 174, No. 3, 400–414 (2011).
N. S. Bondarenko, A. S. Gol’tsev, and V. P. Shevchenko, “Fundamental solutions of the thermoelastic equations for transversely isotropic plates,” Prikl. Mekh., 46, No. 3, 51–60 (2010); English translation: Int. Appl. Mech., 46, No. 3, 287–295 (2010).
A. S. Hol’tsev and V. K. Khyzhnyak, G-Function: Methodical Advice for Learning a “Special Functions” Special Course [in Ukrainian], Donets’k Derzh. Univ., Donets’k (1999).
O. B. Kovalev, A. M. Orishich, V. M. Fomin, and V. B. Shulyat’ev, “Adjoint problems of mechanics of continuous media in gaslaser cutting of metals,” Prikl. Mekh. Tekh. Fiz., 42, No. 6, 106–116 (2001); English translation: J. Appl. Mech. Tech. Phys., 42, No. 6, 1014–1022 (2001).
B. L. Pelekh and M. A. Sukhorol’skii, Contact Problems of the Theory of Anisotropic Elastic Shells [in Russian], Naukova Dumка, Kiev (1980).
Ya. S. Podstrigach and R. N. Shvets, Thermoelasticity of Thin Shells [in Russian], Naukova Dumка, Kiev (1978).
V. K. Khizhnyak and V. P. Shevchenko, Mixed Problems of the Theory of Plates and Shells: Tutorial [in Russian], Donetsk Gos. Univ., Donetsk (1980).
E. Ventsel and Th. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York (2001).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 59, No. 2, pp. 101–108, April–June, 2016.
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Dovbnya, K.M., Dundar, O.D. Construction and Investigation of the Third-Order Approximation to the Solution of the Heat-Conduction Equation for Thin Shallow Shells by Using Legendre Polynomials in the Case of Stationary Heat Exchange. J Math Sci 231, 608–618 (2018). https://doi.org/10.1007/s10958-018-3838-5
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DOI: https://doi.org/10.1007/s10958-018-3838-5