Nonlinear time-fractional diffusion problems, a significant class of parabolic-type problems, appear in various diffusion phenomena that seem extensively in nature. Such physical problems arise in numerous fields, such as phase transition, filtration, biochemistry, and dynamics of biological groups. Because of its massive involvement, its accurate solutions have become a challenging task among researchers. In this framework, this article proposed two operational-based robust iterative spectral schemes for accurate solutions of the nonlinear time-fractional diffusion problems. Temporal and spatial variables are approximated using Vieta-Lucas polynomials, and derivative operators are approximated using novel operational matrices. The approximated solution, novel operational matrices, and uniform collection points convert the problem into a system of nonlinear equations. Here, two robust methods, namely Picard's iterative and Newton's, are incorporated to tackle a nonlinear system of equations. Some problems are considered in authenticating the present methods' accuracy, credibility, and reliability. An inclusive comparative study demonstrates that the proposed computational schemes are effective, accurate, and well-matched to find the numerical solutions to the problems mentioned above. The proposed methods improve the accuracy of numerical solutions from 27 % to 100 % when M>2 as compared to the existing results. The suggested methods' convergence, error bound, and stability are investigated theoretically and numerically.

中文翻译：

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稳健的迭代谱算法，用于时间分数非线性扩散问题的平滑求解和收敛分析


非线性时间分数扩散问题是一类重要的抛物线型问题，出现在自然界中似乎很广泛的各种扩散现象中。此类物理问题出现在许多领域，例如相变、过滤、生物化学和生物群动力学。由于它的大量参与，其准确的解决方案已成为研究人员的一项具有挑战性的任务。在这个框架中，本文提出了两种基于运算的鲁棒迭代谱方案，用于精确求解非线性时间分数扩散问题。时间和空间变量使用 Vieta-Lucas 多项式进行近似，导数运算符使用新颖的运算矩阵进行近似。近似解、新颖的运算矩阵和均匀的收集点将问题转换为非线性方程组。在这里，结合了两种稳健的方法，即 Picard 迭代和 Newton 方法，来处理非线性方程组。在验证现有方法的准确性、可信度和可靠性时，考虑了一些问题。一项包容性的比较研究表明，所提出的计算方案是有效的、准确的，并且匹配良好，可以找到上述问题的数值解。与现有结果相比，当 M>2 时，所提出的方法将数值解的准确率从 27 % 提高到 100 %。从理论和数值上研究了所建议方法的收敛性、误差界限和稳定性。