×

Method of decreasing the order of a partial differential equation by reducing to two ordinary differential equations. (English. Russian original) Zbl 1402.35016

Russ. Math. 62, No. 8, 27-37 (2018); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2018, No. 8, 33-45 (2018).
Summary: Using additional unknown functions and additional boundary conditions in the integral method of heat balance, we obtain approximate analytic solutions to the non-stationary thermal conductivity problem for an infinite solid cylinder that allow to estimate the temperature state practically in the whole time range of the non-stationary process. The thermal conducting process is divided into two stages with respect to time. The initial problem for the partial differential equation is represented in the form of two problems, in which the integration is performed over ordinary differential equations with respect to corresponding additional unknown functions. This method allows to simplify substantially the solving process of the initial problem by reducing it to the sequential solution of two problems, in each of them additional boundary conditions are used.

MSC:

35A24 Methods of ordinary differential equations applied to PDEs
35A35 Theoretical approximation in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Lykov, A. V. Thermal Conductivity Theory (Vysshaya Shkola,Moscow, 1967) [Russian].
[2] Kartashov, E. M. Analytical Methods of Thermal Conductivity of Solids (Vysshaya Shkola, Moscow, 1979) [Russian].
[3] Kantorovich, L. V., Krylov, V. I. Approximate Methods of Higher Analysis (Fizmatgiz, Moscow-Leningrad, 1962; NewYork, Wiley, 1964). · Zbl 0083.35301
[4] Tsoi, P. V. System Calculation Methods for Boundary-Value Problems of Heat and Mass Transfer (Izdatel’stvo MEI,Moscow, 2005) [Russian].
[5] Lykov, A. V., Solution methods for nonlinear equations of non-stationary thermal conductivity, Energetika i Transport, 5, 109-150, (1970)
[6] Goodman, T. R., Application of integral methods to transient nonlinear heat transfer, Advances inHeat Transfer, 1, 51-122, (1964) · Zbl 0126.12705
[7] Kudinov, I. V.; Kudinov, V. A.; Kotova, E. V., Analytic solutions of heat transfer problems based on determining the front of the thermal disturbance, RussianMathematics, 60, 22-34, (2016) · Zbl 1397.80006
[8] Kudinov, V. A., Kudinov, I. V. Analytic Solutions to Parabolic and Hyperbolic Equations of Heat and Mass Transfer (INFRA-M,Moscow, 2013) [Russian]. · Zbl 1291.03035
[9] Stefanyuk, E. V.; Kudinov, V. A., Finding approximate analytic solutions under a mismatch of initial and boundary conditions in problems of heat conduction theory, Russian Mathematics, 54, 55-61, (2010) · Zbl 1197.35291
[10] Kantorovich, L. V., Sur une Méthode de Résolution approcheé d’E´ quations différentielles aux Dériveés partielles, 532-536, (1934) · Zbl 0009.35504
[11] Fedorov, F. M. A Boundary Method of Solution to Applied Problems of Mathematical Physics (Nauka, Novosibirsk, 2000) [Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.