This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

中文翻译：

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B样条小波求解分数阶微分方程的新方法


本文提出了一种用于求解线性和非线性多阶分数微分 (M-OFD) 以及分数偏微分方程 (FPDE) 的突破性方法。该方法涉及使用线性 B 样条 (LB-S) 小波函数以完美的微妙方式构建分数阶导数的运算矩阵。新方法有两个关键特征。首先，它通过将问题转换为一组代数方程来简化问题，这是一个显着的优点。这使得该方法高度准确且可靠。其次，它使用阈值来显着减少线性问题的计算工作量。这可以实现闪电般快速且高效的问题解决。新开发的方案对其误差估计和收敛性进行了彻底的检查，在准确性和效率方面显示出显着的结果。该分析提供了对该方案性能的全面了解，突显了其作为可靠且有效方法的潜力。根据研究结果，很明显，所提出的方法不仅具有出色的精度，而且运行效率极高。