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On a Stable Multiplicative Calculus-Based Hybrid Parallel Scheme for Nonlinear Equations

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  • Mudassir Shams

    (Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100 Bolzano, Italy
    Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44000, Pakistan)

Abstract

Fractional-order nonlinear equation-solving methods are crucial in engineering, where complex system modeling requires great precision and accuracy. Engineers may design more reliable mechanisms, enhance performance, and develop more accurate predictions regarding outcomes across a range of applications where these problems are effectively addressed. This research introduces a novel hybrid multiplicative calculus-based parallel method for solving complex nonlinear models in engineering. To speed up the method’s rate of convergence, we utilize a second-order multiplicative root-finding approach as a corrector in the parallel framework. Using rigorous theoretical analysis, we illustrate how the hybrid parallel technique based on multiplicative calculus achieves a remarkable convergence order of 12, indicating its effectiveness and efficiency in solving complex nonlinear equations. The intrinsic stability and consistency of the approach—when applied to nonlinear situations—are clearly indicated by the symmetry seen in the dynamical planes for various parameter values. The method’s symmetrical behavior indicates that it produces accurate findings under a range of scenarios. Using a dynamical system procedure, the ideal parameter values are systematically analyzed in order to further improve the method’s performance. Implementing the aforementioned parameter values using the parallel approach yields very reliable and consistent outcomes. The method’s effectiveness, reliability, and consistency are evaluated through the analysis of numerous nonlinear engineering problems. The analysis provides a detailed comparison with current techniques, emphasizing the benefits and potential improvements of the novel approach.

Suggested Citation

  • Mudassir Shams, 2024. "On a Stable Multiplicative Calculus-Based Hybrid Parallel Scheme for Nonlinear Equations," Mathematics, MDPI, vol. 12(22), pages 1-29, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3501-:d:1517445
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    References listed on IDEAS

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    2. Naila Rafiq & Saima Akram & Nazir Ahmad Mir & Mudassir Shams, 2020. "Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-20, July.
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    4. Mudassir Shams & Nasreen Kausar & Cuauhtã‰Moc Samaniego & Praveen Agarwal & Shams Forruque Ahmed & Shaher Momani, 2023. "On Efficient Fractional Caputo-Type Simultaneous Scheme For Finding All Roots Of Polynomial Equations With Biomedical Engineering Applications," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(04), pages 1-15.
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