In this research, a two-step bi- and tri-parametric iterative method is developed for solving nonlinear equations employing the Caputo fractional derivative. The theoretical convergence analysis demonstrates that the proposed schemes achieve an order of convergence of $2\tau +1$, where $\tau$ denotes the order of the Caputo fractional operator. In addition, fractal analysis is employed to identify effective initial values that enhance numerical performance. To compare the suggested scheme's efficacy and stability to existing approaches, several nonlinear engineering problems are examined. The numerical results demonstrate that, compared with existing methods, the proposed schemes achieve lower residual errors, higher convergence rates, reduced memory consumption, improved error profiles, and superior computational orders of convergence, thereby making them a more efficient and reliable alternative for solving problems in science and engineering.

The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space $L_2[-1,1],$ while the kernel of Ⅵ was considered as a function of time in the space $C[0,T],T<1$. The existence of a unique solution was discussed in $L_2\left[-1,1\right]\times C\left[0,T\right]$ space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.