Abstract:

This paper extends the Langevin equation on the fractional order to make it time-memory, and a class of fractional-order Langevin equation is solved numerically using the predictor-corrector method. Firstly, the estimated value is obtained by R0 algorithm, then the estimated value is substituted into the R2 algorithm to correct the numerical solution. Finally, the numerical solution of a class of fractional-order Langevin equation by predictor-corrector method is obtained. The error analysis proves that under the condition of $0 < \alpha < 1$ of the equation, the result of predictor-corrector method is $(1+\alpha)$ order convergence. Numerical experiments also show that the numerical solution of the predictor-corrector method is convergent under different values of $\alpha$ and step size $h$.

Key words: Fractional Langevin equation, Predictor-Corrector method, Error analysis, Numerical experiments