Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations

Hari M. Srivastava, Khaled Mohammed Saad and Walid M. Hamanah
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Hari M. Srivastava: Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Khaled Mohammed Saad: Department of Mathematics, College of Sciences and Arts, Najran University, P.O. Box 1988, Najran 66262, Saudi Arabia
Walid M. Hamanah: Interdisciplinary Research Center in Renewable Energy and Power Systems, King Fahd University for Petroleum and Minerals, P.O. Box 5028, Dhahran 31261, Saudi Arabia

Mathematics, 2022, vol. 10, issue 7, 1-13

Abstract: The main objective of this paper is to introduce and study the numerical solutions of the multi-space fractal-fractional Kuramoto-Sivashinsky equation (MSFFKS) and the multi-space fractal-fractional Korteweg-de Vries equation (MSFFKDV). These models are obtained by replacing the classical derivative by the fractal-fractional derivative based upon the generalized Mittag-Leffler kernel. In our investigation, we use the spectral collocation method (SCM) involving the shifted Legendre polynomials (SLPs) in order to reduce the new models to a system of algebraic equations. We then use one of the known numerical methods, the Newton-Raphson method (NRM), for solving the resulting system of the nonlinear algebraic equations. The efficiency and accuracy of the numerical results are validated by calculating the absolute error as well as the residual error. We also present several illustrative examples and graphical representations for the various results which we have derived in this paper.

Keywords: generalized Mittag-Leffler function; multi-space fractal-fractional Kuramoto-Sivashinsky equation; multi-space fractal-fractional; Korteweg-de Vries equation; spectral collocation method involving the shifted legendre polynomials; Newton-Raphson method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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