A numerical method of solving the Cauchy problem for one differential equation with the Caputo fractional derivative

The Cauchy problem for differential equations with fractional derivatives is used in many spheres of science and technology. It was the reason for the development of various methods for its solution, both analytic and approximate ones. The search of an exact solution of differential equations with fractional derivatives by analytic methods is a complex and ineffective task; for this reason, an attempt to solve the considered problem approximately is undertaken in this paper. gated on the segment $[0, T]$. The method of finite differences which is relatively primary to implement is used for the numerical solution. A difference scheme approximating the Cauchy problem with the first order is constructed on a uniform grid. The difference problem is studied for stability and convergence with a fixed value of the function $\alpha(t)$. It is shown that the numerical solution of the problem converges to the exact solution in the first order. Explicit recurrent formulas for the numerical solution are obtained. A computational experiment upon analysis of the numerical solution of the Cauchy problem is carried out. It is shown on the basis of the computational experiment that if we take the average value for $\alpha(t)$, the first order exactness takes place.