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A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

Author

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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

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

Suggested Citation

  • Mudassir Shams & Bruno Carpentieri, 2024. "A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications," Mathematics, MDPI, vol. 12(15), pages 1-33, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2357-:d:1444670
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    References listed on IDEAS

    as
    1. Adilson Elias Xavier & Vinicius Layter Xavier, 2016. "Flying elephants: a general method for solving non-differentiable problems," Journal of Heuristics, Springer, vol. 22(4), pages 649-664, August.
    2. Budzko, Dzmitry & Cordero, Alicia & Torregrosa, Juan R., 2015. "A new family of iterative methods widening areas of convergence," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 405-417.
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