Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation

Roman Parovik,Dmitriy Tverdyi

doi:10.3390/mca26030055

Abstract

The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.

Highlights

IntroductionFractional calculus is widely developed, which is discussed in detail in monographs [1,2,3,4,5], as well as in survey articles, for example [6]
As a rule, questions arise here related to the stability and convergence of numerical methods. We investigate these issues, using an explicit finite-difference scheme in order to find a numerical solution to a nonlinear equation with a fractional variable order derivative
The question of the condition for the appearance of the stiff Cauchy problem (4) deserves a separate study. Another very interesting question deserves attention: how does the variable order of the fractional derivative affect the stability of the explicit finite-difference scheme (6)? It should be noted that the variable order α(t) is included in estimate (10) for the stability of the method

Summary

Introduction

Fractional calculus is widely developed, which is discussed in detail in monographs [1,2,3,4,5], as well as in survey articles, for example [6]. Fractional calculus studies fractional derivatives and integrals, as well as their applications. It is thanks to the applications that fractional calculus is intensively used in physics [7,8], mechanics [9,10,11], biology [12,13], economics [14,15] and other sciences. We will not dwell here on the interpretation of the meaning of the fractional derivative. We refer the reader to other works, such as [16], for discussions on this problem

Results

Discussion

Conclusion