In this work, the distributed-order time fractional version of the Schrödinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss–Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples.

在这项工作中，薛定谔问题的分布阶时间分数版本是通过用这种分数导数替换经典问题中的一阶导数来定义的。Caputo 分数导数用于定义使用的分布分数导数。定义了作为新的基函数族的正交分段雅可比函数。导出了这些函数的 Caputo 分数导数的新公式。一种基于这些分段函数的数值方法以及经典的雅可比多项式和高斯-勒让德正交规则被构造来解决引入的问题。该方法将上述问题转换为可以轻松解决的代数问题。通过求解一些例子对方法的准确性进行了数值检验。