The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both $\mathcal{L}$1 scheme and strong $A$-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong $A$-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $\alpha$-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.
翻译:线性均匀卡普托时间分数平方程式(F-ODEs)的溶液在线性稳定区域和长期衰减率方面已知完全由系数矩阵的偏差值决定。与古典 ODEs 的溶液指数衰减截然不同, F-ODEs 的溶液只会在微量上腐蚀,导致所谓的Mittag-Leffler 偏振度的半线性F-ODEs。这项工作主要用于对数字解决方案的长期行为进行定性分析。通过对Flajolet和Odlyzko(SIAM J. Disar. Math. 3(1990), 216-240)开发的生成函数的奇异性分析,我们能够证明,$降价的低位分数多步法(F-LMMM) 完全可以保持基于线性直线性均匀F-OD-ODR-OF-ODER的数值稳定性。通过不断改进的对离质分数分数的离式平流性平流性平流性平流-O-L-al-I-OD-OD-ID-OD-OD-I-I-OD-ID-OD-OD-OD-IL 的递值变现显示的直流-IL-S-OFD-OD-IF-OF-OF-IFD-OF-S-S-S-S-OD-ODF-OD-OD-OD-SDF-SD-SD-SD-SDFUDF-SDF-SD-S-SDF-SDR-SDF-S-S-SDF-SDF-S-S-SD-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-SDF-S-S-S-S-S-SDF-S-S-SDF-SDF-SD-SD-S-S