In this paper, a modified hybrid B-spline approximation based on spatial regulator tensor network (MHBA-SRTN) is proposed for solving the application and the feasibility problems of fractional calculus-based Burger equation. The main innovation points include: (1) a dual singular kernel based fractional derivative operator is employed on the Burger’s equation with external force term to analyze the solutions and properties under perturbation condition for further realism simulation; (2) a modified hybrid B-spline basis function and its corresponding topological tensor network structure are further proposed to solve the provided Burger’s equation with more favorable approximation accuracy and solution uniqueness; (3) the tensor network transformed the hybrid B-spline construction process into the tensor solving with the introduced spatially regulated tensor network decomposition for a more robust solution procedure; (4) an acceleration approach for decomposition process is introduced to significantly reduce its computational and storage complexity. Extensive experiments are also carried out to verify the performance of MHBA-SRTN.

中文翻译：

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基于非线性分数阶微积分汉堡方程空间调节张量网络的改进混合B样条估计

为了解决基于分数阶微积分的Burger方程的应用和可行性问题，提出了一种基于空间调节张量网络的改进混合B样条逼近（MHBA-SRTN）。主要创新点包括：(1)对带有外力项的Burger方程采用基于双奇异核的分数阶导数算子，分析扰动条件下的解和性质，以进一步进行真实模拟。(2)进一步提出了改进的混合B样条基函数及其相应的拓扑张量网络结构，以更好的逼近精度和解唯一性来求解所提供的Burger方程；(3)张量网络将混合B样条构造过程转化为张量求解，引入空间调节张量网络分解，以获得更鲁棒的求解过程；(4)引入了分解过程的加速方法，显着降低了分解过程的计算和存储复杂度。还进行了大量的实验来验证 MHBA-SRTN 的性能。