Spectral collocation method is proposed to solve Caputo fractional terminal value problem. The main idea of the proposed method is to solve the corresponding nonlinear weakly singular Volterra-Fredholm integral equation. The key step in presented method is to transform the nonlinear integral kernels to linear integral kernels by a functional transformation. Then, the integral terms can be calculated exactly by Gauss quadrature formula. The provided convergence analysis shows that the presented method has spectral convergency. Theoretical results are confirmed by numerical experiments. The presented method makes some improvements to the existed methods.

中文翻译：

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Caputo分数终值问题的光谱搭配方法

为了解决Caputo分数终值问题，提出了一种频谱搭配方法。所提出方法的主要思想是求解相应的非线性弱弱奇异Volterra-Fredholm积分方程。提出的方法的关键步骤是通过功能变换将非线性积分核转换为线性积分核。然后，积分项可以通过高斯正交公式精确计算。所提供的收敛性分析表明，该方法具有频谱收敛性。理论结果通过数值实验得到证实。提出的方法对现有方法进行了一些改进。