A New Model of the Problem with a Fractional Derivative Along the Trajectory of Motion

The article develops the ideas of works [ 1 , 2 ], devoted to the construction and numerical study of mathematical models with ‘‘memory on the flow path’’ for one-dimensional diffusion-convection equations. In this article, we propose a new fractional derivative with respect to the characteristics of the convection operator. For the one-dimensional diffusion-convection equation with the introduced fractional derivative, a finite-difference scheme is constructed based on the Lagrange approach. Stability estimates are proved and accuracy estimates are obtained under the assumption that the input data and the solution of the differential problem are sufficiently smooth functions.