Abstract
The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.
Highlights
AshurovNumerous theoretical and practical studies in the world show that the Riccati equation is of great interest, since it often finds its application in many fields of science, for example, in physics—wave processes in media with inelastic hysteresis and saturation of losses [1], in epidemiology—logistic models, the purpose of which is to determine the time of saturation and recession of the epidemic [2].Saturation processes can have the effect of heredity; this indicates a causal relationship in the dynamics of the process
In the case of the fractional Riccati equation, we prove the theorems of approximation, stability, and convergence for a nonlocal explicit finite difference scheme (EFDS)
Substituting (12) into (31), we obtain a discrete analogue of the Cauchy problem for the fractional Riccati equation, for which the nonlocal implicit finite difference scheme (16)
Summary
Numerous theoretical and practical studies in the world show that the Riccati equation is of great interest, since it often finds its application in many fields of science, for example, in physics—wave processes in media with inelastic hysteresis and saturation of losses [1], in epidemiology—logistic models, the purpose of which is to determine the time of saturation (reaching a plateau) and recession of the epidemic [2]. Little information about numerical methods based on finite difference schemes; no or little comparison of simulation results with real experimental data of processes with saturation; mostly the order of the fractional derivative, is constant, which may produce unacceptable results when describing experimental data; approaches to the numerical solution of the Riccati equation with a fractional variable order derivative are poorly studied. This scientific study is devoted to the elimination of these gaps, the numerical study of the fractional Riccati equation with non-constant coefficients and with a derivative of a fractional variable order of the Gerasimov-Caputo type It addresses questions of convergence and stability of finite-difference schemes. We will evaluate the computational accuracy of numerical methods according to Runge’s rule, as well as compare it with the exact solution
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