A diffusion-convection problem with a fractional derivative along the trajectory of motion

A new mathematical model of the diffusion-convective process with 'memory along the flow path' is proposed. This process is described by a homogeneous one-dimensional Dirichlet initial-boundary value problem with a fractional derivative along the characteristic curve of the convection operator. A finite-difference approximation of the problem is constructed and investigated. The stability estimates for finitedifference schemes are proved. The accuracy estimates are given for the case of sufficiently smooth input data and the solution.