AUTHOR=Aal Mohammad Abdel , Arqub Omar Abu , Maayah Banan 

TITLE=Hilbert solution, iterative algorithms, convergence theoretical results, and error bound for the fractional Langevin model arising in fluids with Caputo’s independent derivative

JOURNAL=Frontiers in Physics

VOLUME=10

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.1072746

DOI=10.3389/fphy.2022.1072746

ISSN=2296-424X

ABSTRACT=<p>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 article, we will attend to a new, accurate, and comprehensive numerical solution to the aforementioned model using the reproducing kernel Hilbert approach. Basically, numerical and exact solutions of the fractional Langevin model are represented using an infinite/finite sum, simultaneously, in the <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>Σ</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>Ξ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> 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, and absolute/relative error graphs that express the dynamic behavior of the fractional parameters <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> are utilized as well. From an analytical and practical point of view, we noticed that the simulation process and the iterative approach are appropriate, easy, and highly efficient tools for solving the studied model. In conclusion, what we have carried out is presented with a set of recommendations and an outlook on the most important literature used.</p>