Extremely low order time-fractional differential equation and application in combustion process
Extremely low order time-fractional differential equation and application in combustion process
复制标题
极低阶时间分数阶微分方程及其在燃烧过程中的应用
DOI：
10.1016/j.cnsns.2018.04.021
复制
发表时间：
2018
影响因子：
3.9
通讯作者：
Xu YF
中科院分区：
文献类型：
--
作者：
Xu Qinwu;Xu Yufeng;Xu YF
分数阶爆破模型，特别是分数阶导数很低的爆破模型，在燃烧过程中起着重要的作用。扩散模型中时间分数阶导数的阶数在相对较高或较低时，从本质上区分了超扩散过程和次扩散过程。本文从理论上证明了爆破现象及其出现的条件。利用微分不等式估计爆破时刻。为了研究爆破点附近的行为，提出了一种基于时间方向自适应有限差分和空间方向高效间断Galerkin方法的混合数值方法.准确计算了爆破时间。在仿真中，我们分析了分数阶导数不同阶数下分数爆破模型的动力学行为。通过固定模型中的其他参数，发现阶数越低，爆破发生的时间越早。我们的结果证实了爆炸燃烧室不能太小的物理事实。
Fractional blow-up model, especially which is of very low order of fractional derivative, plays a significant role in combustion process. The order of time-fractional derivative in diffusion model essentially distinguishes the super-diffusion and sub-diffusion processes when it is relatively high or low accordingly. In this paper, the blow-up phenomenon and condition of its appearance are theoretically proved. The blow-up moment is estimated by using differential inequalities. To numerically study the behavior around blow-up point, a mixed numerical method based on adaptive finite difference on temporal direction and highly effective discontinuous Galerkin method on spatial direction is proposed. The time of blow-up is calculated accurately. In simulation, we analyze the dynamics of fractional blow-up model under different orders of fractional derivative. It is found that the lower the order, the earlier the blow-up comes, by fixing the other parameters in the model. Our results confirm the physical truth that a combustor for explosion cannot be too small.