Research Projects

Fractal interpolation (FI) is known to construct complicated curves starting from simple. Due to its ability to trace irregularity, over the last few decades, FI touched diversified subjects of applied sciences. And thus, in no time, it has become one of the best-fit models for capturing irregular data that arise in physical situations. However, the construction of these curves has an inherent connection with a parameter p ranging between 0 and 1. And we see that with a p-value greater than 1, the computer still generates irregular curves (in fact, the irregularity increases) although the classical theory doesn't support this, resulting in a gap. With an extra scaling parameter, we seek the possibility of a summation of two or more scaling parameters that allow an increase in the range ahead of 1. On the other hand, it has fixed point theory (FPT) as the staunch basis, so any inspection of it would get governed by the Hutchinson-Barnsley (HB) theory of fractals. This notion defines a fractal as a unique fixed point of a finite union of contractive transformations on Hausdorff metric space. Banach contraction condition (BCC), is used to define HB-fractals. But BCC marked the beginning of the FPT and to date, numerous variants of it turned up that can provide additional information to what it gives. Also, these variants embrace a larger class of maps and bring with them a variety of geometrical notions. This fact lured the FPT researchers to replace BCC with its variants to know further about fractals. However, not all of its variants are suitable enough to define fractals, is observed. Pertaining to this, we classified a whole lot of constraints into two- conventional and nonconventional. The concept of FI, fractal function, and their applications have been successful derivatives of HB theory. What made it successful? It is the underlying dynamical system - iterated function system (IFS), as it aids in blending the “fixed point theory” with interpolation. Suitably, every conventional iterated function system (IFS) has delivered fractal, and nonconventional IFSs remained as yet to make a mark until Prithvi and Katiyar tapped into one to successfully prove the assertion of the fractal. It motivated the current project to retrace the literature concerned with nonconventional and revive the theory of fractals in this direction. Hence, the primary objective of this project is to discover suitable nonconventional constraints which can claim the existence of fractals on a complete Hausdorff metric space. As a consequence, the desire to explore the properties of fractals and their mother operator- the union of maps in IFSs, is highly anticipated. To summarize, we have a proposal to revisit the whole of the FPT to study fractals, and thereby the theory of FI, in the light of nonconventional maps. This research carried under this project would fundamentally affect both the FPT and numerical branches of Mathematics towards enriching developments.