Analyzing solution of the fractional Lorenz system and fractional Riccati equation via the modified Laplace power series residual method

Volume 37, Issue 2, pp 154--166 https://dx.doi.org/10.22436/jmcs.037.02.02

Publication Date: September 20, 2024 Submission Date: February 07, 2024 Revision Date: May 09, 2024 Accteptance Date: August 04, 2024

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Authors

A‎. ‎K‎. Alomari - Department of Mathematics‎, ‎Faculty of Sciences, ‎Yarmouk University, ‎Irbid 211-63, ‎Jordan. M‎. Alaroud - Department of Mathematics‎, ‎Faculty of Arts and Science, ‎Amman Arab University, ‎Amman 11953, ‎Jordan. N‎. Tahat - Department of Mathematics‎, ‎Faculty of Science, ‎Hashemite University, ‎Alzarqa 13133, ‎Jordan. M‎. Al-Refai - Department of Mathematics‎, ‎Faculty of Sciences, ‎Yarmouk University, ‎Irbid 211-63, ‎Jordan.

Abstract

‎The series approach is commonly used to obtain approximate analytic solutions for differential equations‎, ‎but it often converges for a short time‎. ‎To address this limitation‎, ‎a new algorithm has been developed that enables the solution to be carried out over a longer period‎. ‎The Laplace residual power series method (LRPSM) is a technique that generates a solution for fractional differential equations in terms of FOPS via simulation generalized Taylors' series in the Laplace space‎. ‎To apply the LRPSM for a long time space‎, ‎a new Modified LRPSM (MLRPSM) algorithm is introduced which divides the time into shorter intervals and applies the LRPSM to each interval‎. ‎The algorithm investigates the continuity of the solution to ensure that the obtained solution for each interval is smoothly connected to the solution for the previous interval‎. ‎The effectiveness of the proposed algorithm is demonstrated through its application to the Riccati equation and the Lorenz chaotic system‎. ‎To comprehend the physical features of studied models‎, ‎the 2D‎, ‎and 3D graphical representations of the acquired results for some parameters had been drawn‎. ‎Especially‎, ‎at the critical value of the fractional derivative‎, ‎which marks the transition of the solution behavior for the Lorenz system from a chaotic to a non-chaotic attractor‎. ‎The efficacy‎, ‎accuracy‎, ‎and feasibility of this technique are verified numerically‎. ‎From this viewpoint‎, ‎the simulations of gained results indicate that the future iterative technique is indeed robust‎, ‎effective‎, ‎and convenient in gaining the approximation solutions over a longer period of a wide range of linear and nonlinear fractional physical problems‎.

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Keywords

• Fractional calculus
• Laplace transform
• residual power series
• multi-stage method
• fractional Riccati equation
• fractional chaotic system

MSC

•  35R11
•  35F25
•  35C10
•  44A10

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