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On Highly Efficient Fractional Numerical Method for Solving Nonlinear Engineering Models

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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)

  • Bruno Carpentieri

    (Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100 Bolzano, Italy)

Abstract

We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3 ς + 5 , according to convergence analysis. Engineering-related numerical test problems are taken into consideration to demonstrate the efficiency and stability of fractional numerical schemes when compared to previously published numerical iterative methods. The newly developed fractional simultaneous approach converges on random starting guess values at random times, demonstrating its global convergence behavior. Although the newly developed method shows global convergent behavior when all starting guess values are distinct, the method diverges otherwise. The total computational time, number of iterations, error graphs and maximum residual error all clearly illustrate the stability and consistency of the developed scheme. The rate of convergence increases as the fractional parameter’s value rises from 0.1 to 1.0.

Suggested Citation

  • Mudassir Shams & Bruno Carpentieri, 2023. "On Highly Efficient Fractional Numerical Method for Solving Nonlinear Engineering Models," Mathematics, MDPI, vol. 11(24), pages 1-30, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4914-:d:1297463
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    References listed on IDEAS

    as
    1. Mudassir Shams & Nasreen Kausar & Naveed Yaqoob & Nayyab Arif & Gezahagne Mulat Addis & Zeljko Stevic, 2023. "Techniques for Finding Analytical Solution of Generalized Fuzzy Differential Equations with Applications," Complexity, Hindawi, vol. 2023, pages 1-31, May.
    2. 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.
    3. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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