In this paper, we propose a robust and simple technique with efficient algorithmic implementation for numerically solving the nonlocal evolution problems. A theta-type (\(\theta \)-type) convolution quadrature rule is derived to approximate the nonlocal integral term in the problem under consideration, such that \(\theta \in (\frac{1}{2},1)\), which remains untreated in the literature. The proposed approaches are based on the \(\theta \) method (\(\frac{1}{2}\le \theta \le 1\)) for the time derivative and the constructed \(\theta \)-type convolution quadrature rule for the fractional integral term. A detailed error analysis of the proposed scheme is provided with respect to the usual convolution kernel and the tempered one. In order to fully discretize our problem, we implement the orthogonal spline collocation (OSC) method with piecewise Hermite bicubic for spatial operators. Stability and error estimates of the proposed \(\theta \)-OSC schemes are discussed. Finally, some numerical experiments are introduced to demonstrate the efficiency of our theoretical findings.

在本文中，我们提出了一种鲁棒且简单的技术，具有高效的算法实现，用于数值解决非局部进化问题。导出theta 型 ( \(\theta \) -type) 卷积求积规则来近似所考虑问题中的非局部积分项，使得\(\theta \in (\frac{1}{2},1 )\)，这在文献中仍未得到处理。所提出的方法基于时间导数的\(\theta \)方法 ( \(\frac{1}{2}\le \theta \le 1\) ) 和构造的\(\theta \)类型分数积分项的卷积求积规则。针对通常的卷积核和经过调节的卷积核，对所提出的方案进行了详细的误差分析。为了完全离散化我们的问题，我们使用分段 Hermite 双三次空间算子实现正交样条配置（OSC）方法。讨论了所提出的\(\theta \) -OSC 方案的稳定性和误差估计。最后，介绍了一些数值实验来证明我们的理论研究结果的有效性。