On the Kernel properties of the integral equation for the model with the essentially loaded heat equation
DOI:
https://doi.org/10.31489/2019No2/105-112Keywords:
thermophysical processes, electric arc, loaded heat equation, boundary value problem, reduced integral equation, kernel of an integral equationAbstract
Mathematical modeling of thermophysical processes in an electric arc of high-current disconnecting apparatuses leads to a boundary value problem for an essentially loaded heat conduction equation. Taking into account the transience of such phenomena, in some cases only a mathematical model is able to give adequate information about their dynamics. The mathematical model in the form of the boundary value problem is reduced to the Volterra integral equation of the second kind, as a result, we have that the solvability of the boundary value problem is equivalent to the solvability of the reduced integral equation. Thus, there is a need to study the reduced integral equation. The results of this study (various representations and properties of the kernel-forming function in general case and the types of the kernel of the integral equation in special cases) are presented in this article. The article is focused at physicists and engineers, as well as scientific researchers engaged in the practical applications of loaded differential equations.
References
"1 Yesbaev A.N., Yessenbayeva G.A., Ramazanov M.I. Investigation of the model for the essentially loaded heat equation. Eurasian Physical Technical Journal. 2019, Vol. 16, No. 1(31), pp. 113 – 120.
Dzhenaliev M.T., Ramazanov M.I. Loaded equations as perturbations of differential equations. Almaty, Publishing house «Gylym», 2010, 334 p.
Prudnikov A.P., Brychkov Yu.A., Marychev O.I. Integrals and series. Vol. 2. Special functions, Moscow, Fizmatlit, 2003, 664 p.
Gradshtein I.S., Ryzhik I.M. Tables of integrals, sums, series and products. Moscow, 1963, 1108 p. [in Russian]
Prudnikov A.P., Brychkov Yu.A., Marychev O.I. Integrals and series. Vol. 1. Elementary functions, Moscow, Fizmatlit, 2002, 632 p.
Yesbayev A.N., Yessenbayeva G.A. On the integral equation of the boundary value problem for the essentially loaded differential heat operator. Bulletin of Karaganda University. Mathematics Series. 2016, No. 3(83), pp. 62 – 69.
Yesbayev A.N., Yessenbayeva G.A., Ramazanov M.I. The research of one boundary value problem for the loaded differential operator of heat conduction. Bulletin of Karaganda University. Mathematics Series. 2013, No. 3(71), pp. 35 – 42.
Polyanin A.D., Manzhirov A.V. Spravochnik po integral'nym uravneniyam: Metody resheniya. Moscow, Publishing House “Factorial Press», 2000, 384 p. [in Russian]
Polyanin A.D., Manzhirov A.V. Spravochnik po integral'nym uravneniyam. Moscow, Fizmatlit, 2003, 608 p. [in Russian]
"
