The work is devoted to the numerical solution of the initial boundary value problem for the heat equation with a fractional Riesz derivative. Explicit and implicit difference schemes are constructed that approximate the boundary value problem for the heat equation with a fractional Riesz derivative with respect to the coordinate. In the case of an explicit difference scheme, a condition is obtained for the time step at which the difference scheme converges. For an implicit difference scheme, a theorem on unconditional convergence is proved. An example of a numerical calculation using an implicit difference scheme is given. It has been established that when passing to a fractional derivative, the process of heat propagation slows down.

This research paper explores the influence of first-order chemical reactions on the sustainable properties of electrically conducting magnetohydrodynamic (MHD) fluids in a vertical channel with the unique characteristics of Jeffrey fluid flow. The mathematical model of MHD flow with Jeffrey fluid and chemical reaction incorporates the impacts of viscous dissipation, Joule heating, and a non-Newtonian fluid model with viscoelastic properties in the flow regions. The governing equations of the flow field were solved using the finite difference method, and the impacts of flow parameters on the flow characteristics were discussed numerically using a graphical representation. It’s revealed that increasing the Jeffrey parameter results in a decline in the velocity field profiles. Also, species concentration field profiles decline with higher values of the destruction chemical reaction parameter. The findings of this study have significant implications for various engineering applications, including energy generation, aerospace engineering, and material processing. Additionally, the inclusion of Jeffrey’s fluid flow introduces a viscoelastic component, enhancing the complexity of the fluid dynamics.