Robust iterative spectral algorithms for smooth solutions of time-fractional nonlinear diffusion problems and convergence analysis

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.