Mudassir Shams

By Mudassir Shams and Bruno Carpentieri

Traditional analytical methods often struggle to handle the inherent complexity of nonlinear fractional problems, highlighting the need for robust numerical techniques. This chapter presents the formulation and stability assessment of novel high-order single-step iterative algorithms developed to solve nonlinear fractional equations arising in scientific and engineering applications. A detailed convergence analysis demonstrates that the proposed schemes achieve a convergence order of δ+1. Through parametric and dynamical analysis, we identify optimal parameter values that ensure stability, consistency, and computational efficiency, with particular attention to sensitivity to the initial guess. The performance of the proposed methods is evaluated against existing techniques in terms of residual error, error plots, computational order, and CPU time on selected real-world engineering problems.

Part of the book: Polynomials

By Mudassir Shams and Bruno Carpentieri

The complexity of the nonlinear eigenvalue problem is too great for traditional analytical methods, particularly in engineering applications where dynamic behaviors and material nonlinearities are crucial. For this reason, this chapter presents a novel hybrid parallel technique based on multiplicative calculus as an effective way to handle scale-invariant processes without dealing with sum divergence issues. It offers improved precision and computational stability, making it especially appropriate for complex systems where ratios—rather than differences—are crucial. The results are verified in the nonlinear dynamical analysis of frame structures, a highly relevant applied engineering problem involving substantial deformations and material nonlinearities. The proposed formulation achieves a convergence order of three. The numerical findings reveal that our method performs better than existing strategies in the literature, with considerable improvements in residual error, computational efficiency, stability, and CPU time. These findings show the method’s potential application in handling real-world engineering challenges such as dynamic structural analysis.

Part of the book: Bridging Eigenvalue Theory and Practice

By Mudassir Shams and Bruno Carpentieri

We propose a new single-step iterative method for solving nonlinear equations in the ring of p-adic integers Zp. The method is derived using a p-adic Taylor expansion, and its convergence is rigorously established in ultrametric spaces under Lipschitz-type conditions. Furthermore, the dynamical behavior of the scheme is explored by generating the basins of attraction over discretized Zp, which reveals fractal structures influenced by the choice of the prime p and the initial approximation. Numerical experiments in Z3 and Z5 confirm the theoretical results and demonstrate that the scheme is stable, sensitive to initial values, and computationally efficient. The observed p-adic fractal patterns resemble classical complex fractals, offering new insights into root-finding dynamics in non-Archimedean fields.

Part of the book: Number Theory