Hilbert solution, iterative algorithms, convergence theoretical results, and error bound for the fractional Langevin model arising in fluids with Caputo’s independent derivative - Aal, Mohammad Abdel; Arqub, Omar Abu; Maayah, Banan (Frontiers in Physics, 2022)
Studying and analyzing the random motion of a particle immersed in a liquid represented in the Langevin fractional model by Caputo's independent derivative is one of the aims of applied physics. In this letter, we will attend to a new, accurate, and comprehensive numerical solution to the mentioned model using the reproducing kernel Hilbert approach. Basically, numerical and exact solutions of the fractional Langevin model were represented using infinite/finite sum, simultaneously, in Σ_2 (Ξ) space. The proof has been sketched for many mathematical theorems such as independence, convergence, error behavior, and completeness of the solution. A sufficient set of tabular results and two-dimensional graphs are shown, as well as, absolute/relative errors graphs that express the dynamic behavior of the fractional parameters (α,β) are utilized too. From an analytical and practical point of view, we noticed that the simulation process and the iterative approach is an appropriate, easy, and highly efficient tools for solving the studied model. In the conclusion, what we have done is presented with a set of recommendations and an outlook on the most important literature used.