APPLICATION OF THE INTEGRAL EQUATION IN THE SOLUTION OF ONE NONLINEAR BOUNDARY VALUE PROBLEM
DOI:
https://doi.org/10.32782/mathematical-modelling/2025-8-1-10Keywords:
spherical area, nonlinear initial-boundary value problem, thermodynamic equilibrium, Green’s function, Hammerstein-type integral equationAbstract
This paper considers a mathematical model of the temperature field of an isotropic spherical area, rotating around one of the symmetry axes at a constant speed in the form of an initial-boundary-value problem for a parabolic-type equation with nonlinear boundary conditions. The purpose of the work is to apply the method of integral equations to the solution of a nonlinear boundary value problem in the spherical domain, to determine the conditions of thermodynamic equilibrium. A periodic problem is considered in one of the spatial coordinates, which is reduced to solving a nonlinear integral equation. The obtained integral equation is reduced to a Fredholm-type equation along the spatial coordinate using the second Green’s formula, and to a Volterra-type equation along the time coordinate. The kernel of the integral equation is found in the form of a Green’s function. The Green’s function is constructed using the properties of the Laplace operator in a spherical coordinate system, as well as the Legendre and Bessel functions of half-integer order. The linear part of the solution function of the problem both inside the sphere and on its surface was found using the properties of the Gamma function. To determine the conditions of thermodynamic equilibrium of a sphere with an unlimited number of its rotations around its axis, a limit transition was applied, as a result of which the conditions of thermodynamic equilibrium were determined. The solution of the nonlinear part of the problem for determining the periodic quas-stationary temperature field was obtained by solving the corresponding nonlinear integral equation of the Hammerstein type. It was also proved that the solution of the original problem, periodic both in time and in the spatial coordinate, does not depend on the initial condition. This condition does not affect the quasi-stationary temperature field of the sphere with an unlimited number of its rotations.
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