Novel iterative methods enhanced by viscosity technique for some fractal generation and applications

This dissertation investigates the complicated structure of fractals, Julia sets, Mandelbrot sets, and biomorphs using the new iteration method based on the viscosity approximation method, which is a well-known fixed point iterative method commonly used to approximate fixed points for non-linear operators. Using this approach, we create the escape criteria for Julia and Mandelbrot sets, which also develop biomorphs for any complex function. In addition, we visualize the sets using the escape time method and the proposed iteration to develop algorithms to create Julia sets, Mandelbrot sets, and biomorphs. Then, using graphical and numerical experiments, we investigate how the form of the generated sets varies according to the iteration settings. The examples demonstrate that this transition may be complicated, resulting in various shapes. Moreover, using a modified viscosity approximation approach, we propose a general iterative strategy for introducing a condition that fulfils the escape condition of all complexvalued functions in a novel class that includes multiple types of complex-valued functions in the literature review. We provide a theory that establishes the escape criteria, or the condition when a sequence diverges to infinity, for the given complex functions.Furthermore, we establish corollaries that fulfill the suggested iteration approach for the provided theorem, as demonstrated using examples and visual representations. We also generate Julia and Mandelbrot sets based on escape conditions, with numerical examples provided in Mathematica software. We investigate the relationships between iterations using three numerical measures: average escape time, non-escaping region index, and fractal creation time.