A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

Laila A Al-Essa; Mati Ur Rahman

doi:10.1038/s41598-024-57389-0

A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

Sci Rep. 2024 Mar 24;14(1):6990. doi: 10.1038/s41598-024-57389-0.

Authors

Laila A Al-Essa¹, Mati Ur Rahman^{2

3}

Affiliations

¹ Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O.Box 84428, 11671, Riyadh, Saudi Arabia.
² School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, Jiangsu, People's Republic of China. MatiUr.Rahman@lau.edu.lb.
³ Department of computer science and mathematics, Lebanese American university, Beirut , Lebanon. MatiUr.Rahman@lau.edu.lb.

PMID: 38523147
DOI: 10.1038/s41598-024-57389-0

Abstract

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution's existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.

Keywords: Fixed point; Fractal-fractional operator; Kawahara equation; Unique solution.