Research on the numerical solution and dynamic properties of nonlinear fractional differential equations

Abstract

Fractional calculus is an important branch of mathematical analysis, which is specialized in the study of the mathematical properties and applications of arbitrary order integral and differential, and is the extension of the traditional integral calculus. At present, fractional integral and derivative operators are mainly used to calculate fractional calculus, among which the most famous ones are Riemann-Liouville fractional integral and derivative, Caputo fractional derivative, Grümwald-Letnikov fractional integral and derivative, etc. At present, the numerical algorithm of finite difference scheme is mainly used to solve the approximate solution of the equation, to solve the fractional differential equation. Through the finite difference of time fractional order or space fractional order, the approximate solution of the equation is obtained, and the stability, convergence and compatibility of the scheme are checked, and the convergence order and estimation error are calculated. At present, the theory and method of nonlinear fractional differential equation are widely used in the study of various intermediate processes and critical phenomena in finance, physics and mechanics, which can better fit some natural physical processes and dynamic system processes.