Abstract This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable. In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.

中文翻译：

---

Caputo型时间分数修正异常子扩散方程的半解析方法

摘要 本文致力于求解时间分数修正异常子扩散方程（FMASDE）的一种新的半解析算法。在该方法中，首先使用切比雪夫搭配程序将主要问题简化为已知初始值条件下的分数阶常微分方程（FODE）系统。之后，为了求解该系统，定义了一些辅助初值问题。接下来，我们为这些问题找到一些特定解的最优线性组合，最后我们使用这种线性组合来构造主要问题的半解析近似解。为了证明新方法的收敛性，进行了详细的残差分析。研究了一些测试问题，以证明所提出方法的可靠性和准确性。除了，收敛阶指标对所有测试问题进行评估，并与其他方法进行比较。此外，我们计算的数值结果与文献中其他数值方案的报告结果之间的比较表明，所提出的技术更加精确和可靠。综上所述，该方法的优点是：精度高、易于编程、实验收敛阶数高、求解其他类型的分数阶微分方程。