Analysis of Fast Convergent Iterative Scheme with Fractal Generation

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form T(u)=ua−ξu2+ru+sinρσforallu∈C and a∈N∖{1}, ξ∈C, r,ρ∈C∖{0} are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique.