Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation




nonlinear thermal conductivity, positive solution, Green’s function, two-sided iterative method, equation with isotonic operator


The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes.

Author Biographies

Nadiia Gybkina, Kharkiv National University of Radio Electronics, Kharkiv

Nadiia V. Gybkina,

Candidate of Technical Sciences (Ph.D.), an associate professor at the Department of Applied Mathematics of Kharkiv National University of Radio Electronics, Kharkiv, Ukraine.

Maxim Sidorov, Kharkiv National University of Radio Electronics, Kharkiv

Maxim V. Sidorov,

associate professor, Doctor of Physical and Mathematical Sciences, a professor at the Department of Applied Mathematics of Kharkiv National University of Radio Electronics, Kharkiv, Ukraine.

Kostiantyn Vasylyshyn, Kharkiv National University of Radio Electronics, Kharkiv

Kostiantyn V. Vasylyshyn,

a student at the Faculty of Information and Analytical Technologies and Management of Kharkiv National University of Radio Electronics, Kharkiv, Ukraine.


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Mathematical methods, models, problems and technologies for complex systems research