The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method

Abstract

Abstract In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L ⁢ 1 {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1 ) {\alpha\in[\frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted L ⁢ 1 {L1} method is proposed and the region of optimal convergence order is larger. For the case t > s {t>s} , stability and min ⁡ { 2 ⁢ α , 1 } {\min\{2\alpha,1\}} order convergence of the fitted L ⁢ 1 {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.