Abstract In this paper, the space-time Jacobi spectral collocation method (JSC Method) is used to solve the time-fractional nonlinear Schr o ¨ dinger equations subject to the appropriate initial and boundary conditions. At first, the considered problem is transformed into the associated system of nonlinear Volterra integro partial differential equations (PDEs) with weakly singular kernels by the definition and related properties of fractional derivative and integral operators. Therefore, by collocating the associated system of integro-PDEs in both of the space and time variables together with approximating the existing integral in the equation using the Jacobi-Gauss-Type quadrature formula, then the problem is reduced to a set of nonlinear algebraic equations. We can consider solving the system by some robust iterative solvers. In order to support the convergence of the proposed method, we provided some numerical examples and calculated their L∞ norm and weighted L2 norm at the end of the article.

中文翻译：

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求解时间分数阶薛定谔方程的时空雅可比谱搭配方法的收敛性分析

摘要 本文采用时空雅可比谱搭配法（JSC Method）求解具有适当初始和边界条件的时间分数非线性Schr ö ¨ dinger方程。首先，通过分数阶导数和积分算子的定义和相关性质，将所考虑的问题转化为具有弱奇异核的非线性Volterra积分偏微分方程(PDE)的关联系统。因此，通过在空间和时间变量中搭配相关的 integro-PDEs 系统，并使用 Jacobi-Gauss-Type 正交公式逼近方程中的现有积分，然后将问题简化为一组非线性代数方程. 我们可以考虑通过一些鲁棒的迭代求解器来求解系统。