Search

The paper presents a computational algorithm for solving an optimal control problem formulated in terms of mean-field game theory. The equilibrium of the mean field leads to a coupled system of two parabolic partial differential equations: Fokker-Plank-Kolmogorov and Hamilton-Jacobi-Bellman ones. We extend the conventional model by incorporating a mechanism for external (crisis) influence on the behavior of agents. The article is devoted to the discrete approximation of such a differential statement. The constructed algorithm is implemented to the task of heating insulation of houses. [ABSTRACT FROM AUTHOR]

The paper presents a new mathematical model of the convection-diffusion process with 'memory along the flow path'. It is described by one-dimensional initial-boundary value problem with a fractional derivative along the characteristic curve of convection operator. The constructed model satisfies the local and global conservation laws. The finite-difference approximation of the problem is constructed on the base of Lagrange approach. The stability and discrete conservation laws are proven for algorithmic realization of this approximation. [ABSTRACT FROM AUTHOR]

We present the semi-Lagrangian approximation of transfer operator for three-dimensional convection-diffusion problem with corresponding initial and boundary conditions. This problem describes, for instance, a transfer of a substance with diffusion. To construct numerical method, we decompose operator of this equation into two parts. The first one is the transfer operator. The second part is the elliptic diffusion terms. To approximate the first part, we use conservative semi-Lagrangian approximation connecting two integrals of solution at neighboring time levels. To compute integral at the previous time level, we approximate integral domain by 48 tetrahedrons and interpolate solution by trilinear functions. The second part is approximated by conventional finite differences. Finally, we combine approximations of two parts to get a system of linear equations at each time level. Matrix of this system at each time level is the symmetric M-matrix. The proposed approximation has the first-order convergence that is confirmed by computational experiments. [ABSTRACT FROM AUTHOR]