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Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

Khursheed J. Ansari1, Mustafa Inc2,3,4,*, K. H. Mahmoud5,*, Eiman6
1 Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
2 Department of Computer Engineering, Biruni University, Istanbul, 34010, Turkey
3 Department of Mathematics, Science Faculty, Firat University, Elazig, 23119, Turkey
4 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
5 Department of Physics, College of Khurma University College, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia
6 Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, 18000, Pakistan
* Corresponding Authors: Mustafa Inc. Email: minc@firat.edu.tr; K. H. Mahmoud. Email: k.hussein@tu.edu.sa
(This article belongs to this Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2022.022971

Received 02 April 2022; Accepted 06 June 2022; Published online 06 September 2022


In this article, we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative (CFFD). The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii. Also, we enriched our work by establishing a stable result based on the Ulam-Hyers (U-H) concept. Also, the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method. We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders. The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier. Also, the suggested method results for the proposed system under the mentioned derivative are new. Furthermore, the adopted technique has some useful features, such as the lack of prior discrimination required by wavelet methods. our proposed method does not depend on auxiliary parameters like the homotopy method, which controls the method. Our proposed method is rapidly convergent and, in most cases, it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.


Lorenz system; CFFD; fixed point approach; approximate solution
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