#### 519.6 Conservative difference schemes for the optimal selection of the location of heat sources in the rod

##### Khayitkulov B. K. (National University of Uzbekistan)

doi: 10.18698/2309-3684-2020-8598

In this paper, we have developed a method and an algorithm for solving the problem of the optimal selection of the density of heat sources on the rod in such a way that the temperature inside the considered area is within the specified limits is developed. In this case, the heat sources must provide the specified temperature regime of the minimum total power and temperature in the specified temperature corridor. Conservative finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method for constructing conservative difference schemes for solving the heat conduction equation with variable coefficients, a brief description of the developed software application for constructing computational grids and solving equations is presented. A new method for the numerical solution of non-stationary problems of optimal selection of heat sources in a rod is proposed and substantiated. A software application has been created for carrying out numerical experiments to solve the problem. The description of the based algorithm and the results of numerical experiments are given.

Хайиткулов Б.Х. Консервативные разностные схемы по оптимальному выбору местоположения источников тепла в стержне. Математическое моделирование и численные методы, 2020, № 3, с. 85–98.

#### 519.6 Mathematical modeling of the non-stationary convection-diffusion problem on the optimal choice of heat sources location

##### Khayitkulov B. K. (National University of Uzbekistan)

doi: 10.18698/2309-3684-2023-1-3242

This work is devoted to the numerical solution of the non-stationary problem of optimal placement of heat sources of minimum power. The statement of the problem requires the simultaneous fulfillment of two conditions. The first condition is to ensure that the temperature is within the limits of minimum and maximum temperatures due to the optimal placement of heat sources with a minimum power in the parallelepiped. The second condition is that the total power of the heat sources used for heating is minimal. This problem was studied under stationary conditions in the works of other scientists. However, the problem was not considered in the non-stationary case. Since it is difficult to find a continuous solution to the boundary value problem, we are looking for a numerical solution to the problem. It is difficult to find an integral operator with a continuous kernel (Green's function). The numerical value of the Green's function is found in the form of a matrix. A new algorithm for the numerical solution of a non-stationary optimal control problem for the placement of heat sources with a minimum power in processes described by parabolic partial differential equations is proposed. A new technique for numerical solution is proposed. A mathematical and numerical model of the processes described by the convection-diffusion equation given for the first boundary value problem is constructed. The boundary value problem is studied for the three-dimensional case. An implicit finite difference scheme was used to solve the problem numerically. According to this scheme, a system of difference equations was created. The formed system of difference equations is reduced to a linear programming problem. The problem of linear programming is solved using the M-method. For each time value, a linear programming problem is solved. A new approach to the numerical solution of problems is proposed. A general block diagram of the algorithm for solving the non-stationary problem of optimal control of the placement of heat sources with a minimum power is given. An algorithm and software for the numerical solution of the problem have been developed. A brief description of the software is given. On specific examples, it is shown that the numerical solution of the boundary value problem is within the specified limits, the sum of optimally placed heat sources with a minimum power gives a minimum to the functional. The results of the computational experiment are visualized

Хайиткулов Б.Х. Математическое моделирование нестационарной задачи конвекции–диффузии об оптимальном выборе местоположения источников тепла. Математическое моделирование и численные методы, 2023, No 1, с. 32–42.