This study proposes an efficient and stable technique based on new hybrid B-spline (HB-spline) functions for the numerical treatment of the Caputo time fractional nonlinear Burgers’ (TFNB) equation. The time derivative is discretized using the definition of the Caputo derivative, whereas HB-spline functions are used to discretize the spatial derivatives. The Rubin–Graves technique is used to linearize the nonlinear terms. The performance and efficacy of the established method are tested using three examples. The graphical results represent the smoothness between numerical and exact solutions. The absolute errors are very low as $$1{0}^{-4}$$ and $$1{0}^{-5}$$ . The convergence rate shows that the proposed method is second-order accurate in space. The proposed method provides better results than the methods available in the literature. The method yields highly accurate results and can handle large-scale problems, which is the novelty of the present work.

中文翻译：

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Caputo时间分数阶非线性Burgers方程的混合B样条配置技术

本研究提出了一种基于新型混合 B 样条 (HB 样条) 函数的高效稳定技术，用于卡普托时间分数非线性 Burgers (TFNB) 方程的数值处理。使用 Caputo 导数的定义来离散时间导数，而使用 HB 样条函数来离散空间导数。Rubin-Graves 技术用于线性化非线性项。使用三个示例测试了所建立方法的性能和功效。图形结果表示数值解和精确解之间的平滑度。绝对误差非常低，为 $$1{0}^{-4}$$ 和 $$1{0}^{-5}$$ 。收敛速度表明该方法在空间上具有二阶精度。所提出的方法提供了比文献中可用的方法更好的结果。该方法产生高度准确的结果，并且可以处理大规模问题，这是当前工作的新颖之处。