Research Projects

Fractal geometry has emerged as an extension of Euclidean geometry to investigate the naturally existing irregular patterns, rough functions and surfaces. In the field of fractal analysis and approximation theory, fractal interpolation is one of the most popular research themes. Various studies in the development and analysis of fractal functions include: the construction of types of fractal functions and fractal splines, application of fractional calculus methods, identification of parameters, investigation of approximation properties of the fractal operator. In all the above-mentioned studies, Banach fixed point theorem has been used as the primary tool for generating and analysing the various aspects of fractal functions. Recently, a non-linear fractal interpolation function has been constructed using the Matkowski and the Raktoch fixed point theorems. In addition, fractals have been generated using the Kannan contractions and Reich contractions. However, there is no research reported to explore the generation of fractal perturbations and its approximation properties involving contractions other than Banach contraction. Also, though fractal splines exist in the literature, its fractional calculus methods have not been discussed so far despite the study of fractal functions. Hence, to bridge these two research gaps, the present proposal targets to employ different contractions for generating perturbed fractal functions and examine its elementary properties. In addition, the proposal plans to identify and optimize the scaling parameters of the fractal splines by applying the Caputo fractional derivative, for the best approximation of prescribed continuous function. Moreover, the Caputo fractional derivative of the fractal splines can be employed to reconstruct the chaotic attractors associated with the dynamical systems to reveal their hidden self-similar nature.