在许多实际应用背景中，分数阶算子的奇性导致了分数阶模型解的奇性；根据经典的离散Gronwall型不等式的形式，Dixon和McKee提出了离散分数阶Gronwall型不等式，这种不等式目前还没有被用于非线性分数阶偏微分方程的稳定性和收敛性证明。针对以上两个实际数值问题，我们利用修正方法和多区域谱配点法，处理非线性分数阶模型的奇性问题，并给出相关理论分析；将上述离散分数阶Gronwall型不等式，应用于非线性时间分数阶偏微分方程的理论分析，并导出不等式关于时间差分方法的统一的分析框架。. 随着微机电系统的发展，微纳尺度管道中的流体技术研究对生物、医学、电学和传热传质等相关的应用领域具有指导意义。在直流电场和交变电场驱动下，建立非线性分数阶粘弹性流体微流动的动力学模型，利用高效谱方法发展模型的数值模拟技术，给出理论证明，探究模型的奇性问题，并进一步揭示微流体的流动机理。

In many practical applications, the solutions usually have singularity for fractional models, which is caused by the singularity of fractional operators. According to the form of classical discrete Gronwall type inequality, Dixon and McKee proposed discrete fractional Gronwall type inequality, which has not been used to prove the stability and convergence of nonlinear fractional partial differential equations. For the above two practical numerical problems, we develop the correction method and the multi-domain spectral collocation method to deal with the singularity, and present the corresponding theoretical analysis. The above-mentioned discrete fractional Gronwall type inequality is applied to the theoretical analysis of the nonlinear time-fractional partial differential equations, and the unified analysis framework for time difference methods is derived.. With the development of MEMS, the research of fluid technology in micro and nano scale pipes is of great significance in the fields of biology, medicine, electricity, heat and mass transfer. Driven by the direct current electric field and alternating electric field, the dynamic models of nonlinear fractional viscoelastic fluid micro flow are established. The numerical simulation technology of the models is developed by using the high-efficiency spectral methods. The theoretical proof is given, and the singularity of the models is explored. We further reveal the flow mechanism of the microfluid flow.