Differential and Fourier Analysis
Global problems of hyperbolic equations are addressed using new metrics and their associated pseudo-differential calculus. New identities for partial fractions of cyclotomic polynomials are derived. The identities are applied to several problems in the area of partition of numbers. Further, using Fourier analysis, the Fourier-Dedekind sums and the associated reciprocity theorems are generalized.
Studies on controllability of various kinds of differential equations with instantaneous as well as non-instantaneous impulses in abstract spaces is carried out. The research focus is to get insight into the different types of controllability, existence, and stability of solutions of fractional differential equations. For example, existence and uniqueness of solutions for Caputo-Hadamard fractional differential equation with impulsive boundary conditions is attempted.
Studies related to qualitative properties of solutions of generalized higher order systems ordinary differential equations is in progress. Conditions guaranteeing solutions of such systems with certain asymptotic properties are being worked out.
The classical cubic spline does not give appropriate solutions if the non-homogeneous differential equation involves a continuous function that is not differentiable. Since the classical cubic spline is a particular case of fractal spline, a method to solve the two-point BVPs using the cubic spline FIFs through moments is proposed.
Fractal interpolation is a modern and advance tool to analyze various scientific and natural data that are non-smooth in nature. We proposed a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal versions of the classical rational cubic trigonometric polynomial spline to analyze different shape preserving properties of a given data.