SISTEMA DE ANÁLISIS, MODELIZACIÓN Y OPTIMIZACIÓN COMPUTER SIMULATION SYSTEM FOR THE NUMERICAL SOLUTION OF THE HEAT EQUATION WITH A POWER-LAW NONLINEARITY BY MESHLESS METHOD

The nonlinear parabolic partial differential equations of the second order are the basis of many mathematical models used in physics, mechanics, biology, chemistry, and ecology. For example, the nonlinear heat equation describes the processes of electron and ion thermal conductivity in a plasma, of adiabatic filtration of gases and liquids in porous media, blood flow in capillaries, diffusion of neutrons and alpha particles in reactor materials, chemical kinetics and biological activity. Nonlinear heat conduction processes were first studied by Zel’dovich and Kompaneets The authors considered the process of heat transfer using the mechanism of radiative heat conduction from an instantaneous point source for a one-dimensional problem. The solution to this problem is obtained in an analytical form. The heat equation with a power-law nonlinearity is especially common among the equations of this type. The universal character of this equation makes it possible to assert that the numerical solution of boundary-value problems, which are described by the heat equation withs a power-law nonlinearity, a relevant research. Authors computer simulation the numerical of the one-dimensional nonstationary heat equation a power-law nonlinearity


UKRAINE
The nonlinear parabolic partial differential equations of the second order are the basis of many mathematical models used in physics, mechanics, biology, chemistry, and ecology. For example, the nonlinear heat equation describes the processes of electron and ion thermal conductivity in a plasma, of adiabatic filtration of gases and liquids in porous media, blood flow in capillaries, diffusion of neutrons and alpha particles in reactor materials, chemical kinetics and biological activity.
Nonlinear heat conduction processes were first studied by Zel'dovich and Kompaneets [1]. The authors considered the process of heat transfer using the mechanism of radiative heat conduction from an instantaneous point source for a one-dimensional problem. The solution to this problem is obtained in an analytical form.
The heat equation with a power-law nonlinearity is especially common among the equations of this type. The universal character of this equation makes it possible to assert that the numerical solution of boundary-value problems, which are described by the heat equation withs a power-law nonlinearity, is a relevant research.
Authors have developed the computer simulation system for the numerical solution of the one-dimensional nonstationary heat equation with a power-law nonlinearity by meshless method [2].

The initial condition is
The boundary conditions are By substitution ( , ) = 1/ ( , ) in equation (1), we obtain the equation (2) with initial condition (3) and boundary conditions: Using the forward Euler method for time discretization [3], we obtain where: number of collocation points, = ‖ − ‖ -Euclidean distance between points, unknown coefficients to be determined. (4), we obtain the system of linear algebraic equations, which can be solved for unknown coefficients . By substituting the values of in (5), approximate solution of the equation (2) at -th step can be obtained.

Substituting (5) into
This iterative method was implemented in the created computer simulation system.
The interface of the computer simulation system is shown in Fig. 1.

Fig. 1. Interface of the computer simulation system
At the top of the program is tab Choice RBF. The Choice RBF tab allows select the type of radial basis function. The computer simulation system allows is used the following radial basis functions: Gaussian, multiquadric, inverse quadratic, and inverse multiquadric. The computer simulation system allows setting the initial condition and boundary conditions of the boundary-value problem. In the computer simulation system, it is possible to set such parameters of the solution as the exponent in the nonlinear heat equation, the coefficient of thermal conductivity, the density, the specific heat at constant pressure, the size of the domain of the boundary-value problem, the distance between interpolation nodes, the time interval of the nonstationary boundary-value problem, the time step size, and the shape parameter in radial basis function.
The approximate solution of the one-dimensional nonstationary heat equation with power-law nonlinearity in computer simulation system is visualized as threedimensional surface (Fig. 1). The computer simulation system allows visualization of the numerical solution at selected time steps as three-dimensional plots.