The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

中文翻译：

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基于分数插值理论和改进神经网络基础的分布式阶次二维分数反应扩散方程非光滑解的无网格方法


本研究论文的主要目的是提出一种新颖且有效的无网格数值方法，用于求解任意域上具有分布式顺序的二维时间分数反应扩散系统。应用高斯-勒让德求积公式来离散化分布阶导数积分。我们建立了分段抛物型分数插值理论，并在其帮助下，所提出的方法可以熟练地求解方程组的非光滑解，并更准确地逼近Caputo分数阶导数。这种基于改进的神经网络基的无网格方法结合了神经网络的高精度逼近优势和强大的表达能力，可以构造任意域上的基函数集，大大减少了计算消耗。基于神经网络构建的碱基允许选择12个碱基数，以达到相当于1000个普通碱基的近似值。对无网格方法的误差和收敛阶数进行了理论分析。数值例子验证了无网格数值方法的高精度和能力。