This work devotes to developing a systematic and convenient approach based on the celebrated convolution quadrature theory to design and analyze difference formulas for fractional calculus at an arbitrary shifted point \(x_{n-\theta }\). The developed theory, called shifted convolution quadrature (SCQ), covers most difference formulas from the aspects of characterizing the formation of related generating functions which are convergent with integer orders. For stability reasons, the theoretical determination of shifted parameter \(\theta \) is provided to fill the gap in which the choice of \(\theta \) depends heavily on experiments particularly for non-integer order derivatives. Further, to discuss the effects of \(\theta \) on A(\(\delta \))-stability, stability regions for several generalized popular formulas within SCQ are examined which are crucial to developing robust numerical schemes. Some numerical tests are also considered to demonstrate the necessity of introducing \(\theta \) for theoretical and practical purposes.

这项工作致力于开发一种基于著名的卷积求积理论的系统且方便的方法来设计和分析分数阶微积分在任意位移点\(x_{n-\theta }\)处的差分公式。发展起来的理论，称为移位卷积正交（SCQ），从表征相关生成函数的形成方面涵盖了大多数差分公式，这些生成函数是整数阶收敛的。出于稳定性原因，提供了移位参数\(\theta \)的理论确定以填补\(\theta \)的选择在很大程度上依赖于实验的空白，特别是对于非整数阶导数。进一步讨论\(\theta \)对 A(\(\delta \) )-稳定性，SCQ 中几个通用公式的稳定性区域被检查，这对于开发稳健的数值方案至关重要。一些数值测试也被认为是为了证明出于理论和实践目的引入\(\theta \)的必要性。