Abstract:Background. The object of study in this work is the Cauchy problem for an ordinary differential equation with the Riemann - Liouville fractional derivative on an interval $[0,T]$. A distinctive feature of the problem is that the order is a variable function $ \alpha = \alpha (t)$
that depends on time and satisfies the condition $0< \alpha (t) <1$. The purpose of the study is to construct a numerical method for solving the designated Cauchy problem. Materials and methods. For a numerical solution, the finite difference method is used, with the help of which the transition from a continuous region to a discrete one is carried out. A difference approximation of the Riemann - Liouville fractional derivative is used based on the definition of the Grunwald - Letnikov fractional derivative. Results. A difference scheme is constructed that approximates the original problem with order $2- \alpha (t)$. The convergence and stability of the difference solution is proven. A computational experiment was also carried out for various functions $ \alpha (t)$. Conclusions. The performed computational experiment confirms the convergence of the proposed method.
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