This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal direction of the considered model using the Legendre orthogonal polynomials. Moreover, to archive a full discretization approach a type of nonlocal method has been applied that is known as the nonlocal peridynamic differential operator (PDDO). The PDDO is based on the concept of peridynamic (PD) interactions by proposing the PD functions orthogonal to each term in the Taylor Series Expansion (TSE) of a field variable. The PDDO for numerical integration uses the vicinity of each point (referred to as the horizon, which does not need background mesh). The PDDO is exclusively described in terms of integration (summation) throughout the interaction domain. As a result, it is unsusceptible to singularities caused by discontinuities. It does, however, need the creation of PD functions at each node. We numerically investigate the stability, the convergence of the scheme, which verifies the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.

本研究主要研究获得时间分数扩散方程数值解的新技术。分数导数项以拉格朗日运算意义表示。首先，我们使用勒让德正交多项式描述所考虑模型的时间方向。此外，为了归档完全离散化方法，已经应用了一种非局部方法，称为非局部近场动力学微分算子 (PDDO)。PDDO 基于近场动力学 (PD) 相互作用的概念，通过提出与场变量的泰勒级数展开 (TSE) 中的每个项正交的 PD 函数。用于数值积分的 PDDO 使用每个点的附近（称为地平线，不需要背景网格）。PDDO 专门根据整个交互域的集成（求和）来描述。因此，它不会受到由不连续性引起的奇点的影响。但是，它确实需要在每个节点上创建 PD 功能。我们对方案的稳定性、收敛性进行了数值研究，验证了所提方法的有效性。数值结果表明了所提出方法的简单性和准确性。