Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))

Abstract

This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.

Excision theory in the dihedral and reflexive (co)homology of algebras

“…The results of the excision theorem of the dihedral cohomology equipped with the results of Intissar (2020) and Kostikov and Romanenkov (2020). Also, our results can introduce this application in a new form.…”

“…We can apply this theorem in the operator algebra in the next work and apply this result in our life working as by Intissar (2020) to improve results. Our result can be equipped with the result by Kostikov and Romanenkov (2020) and can improve it by using our result.…”

Sensitivity Analysis on Thermal Performance of Gas Heater with Finned and Finless Tubes using Characteristics-based Method

“…Zhang et al [13] studied a gas engine-driven heat pump (GEHP) performance experimentally for space heating and cooling and investigated the effect of critical parameters on system performance under both cooling and heating modes. Kostikov and Romanenkov [14] performed an estimating the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. A variety of longitudinal and peripheral fins are used to increase the heat transfer of free convection.…”

Stability for stochastic neutral integro-differential equations with infinite delay and Poisson jumps

“…Research on stochastic differential equations with delay has received attention over the last few decades because of their appropriateness to describe physical systems subject to delays, such as the ones found in biology, medicine, epidemiology, chemistry, physics, and economics (see Helge et al, 2010;Intissar., 2020;Kostikov & Romanenkov, 2020;Mao, 2007;Trung, 2020 for a brief overview). The qualitative and quantitative properties of solutions of stochastic differential equations with delay, such as the existence, uniqueness, controllability, and stability, have been considered by several authors (see Bouzahir et al, 2017;Dieye et al, 2017;Diop et al, 2014;Taniguchi et al, 2002;Zouine et al, 2020).…”