On a numerical method for solving a Cauchy-type problem for an ordinary differential equation with a fractional derivative by Caputo – Fabrizio

Background. Fractional integro-differentiation is currently used in numerous studies in various fields of natural science, such as thermodynamics, radiophysics, biophysics, biology, aerodynamics, viscoelasticity, and electrical circuit theory. Along with practical applications, fundamental methods in the theory of fractional differential equations and their applications are also currently being developed, and new definitions of fractional derivatives are emerging. One new definition of the fractional derivative was introduced relatively recently by M. Caputo and M. Fabrizio. This new definition of the derivative lacks a singularity in the kernel, which, according to the authors, allows it to effectively describe the memory effect and is also capable of representing material inhomogeneities and structures in various cases, which are physically symbolized by differences or variations in the mean. Aim. The aim of the work is to develop numerical algorithms for solving a Cauchy-type problem for a system of ordinary differential equations with fractional derivatives of Caputo – Fabrizio order and computational experiments to analyze numerical solutions. Methods. The mathematical apparatus of fractional derivatives and integrals is applied. A numerical method for solving a Cauchy-type problem for a system of differential equations with the Caputo – Fabrizio fractional derivative is employed. Results. This paper develops numerical algorithms for solving a Cauchy-type problem for a system of ordinary differential equations with fractional Caputo – Fabrizio derivatives. Using second-order finite-difference approximation, a numerical method for solving a Cauchy-type problem for a system of differential equations with fractional Caputo – Fabrizio derivatives is constructed, and a convergence theorem for the numerical method is proven. Computational experiments are conducted to analyze the numerical solution. Conclusion. In this paper, a numerical algorithm for solving a Cauchy-type problem for a system of ordinary differential equations with Caputo – Fabrizio fractional derivatives is developed, and convergence conditions for the numerical method are obtained. The algorithm’s performance is verified on a test problem. We have shown that the error coincides with the accuracy of the method. A fractional Lorenz oscillator with a memory effect through a Caputo – Fabrizio fractional derivative is studied as a nonlinear dynamic system with chaotic dynamics. A Python program is developed based on the algorithm for numerically solving the Cauchy problem for a differential equation with a Caputo – Fabrizio fractional derivative. Phase trajectories are plotted for various values of the system parameters. Topological changes in the phase plane upon transition to fractional derivatives are established.