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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]

The paper presents a computational algorithm for solving problems formulated in terms of Mean Field Game theory with diffusion expressed by fractional derivative. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: modified Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman equations with the highest derivative in space of fractional order less than 2. The article is devoted to the discrete approximation of these equations and the computational algorithm for solving discrete analogues. [ABSTRACT FROM AUTHOR]