A Study of certain fractional-order boundary value problems on non-regular domains

Abstract

The objective of this thesis is to investigate fractional-order boundary value problems in non-regular domains by examining the existence and uniqueness of solutions for various types of abstract differential equations involving fractional operators. The study begins with an analysis of three-dimensional fourth-order differential equations incorporating fractional powers of the negative Laplace operator under Cauchy-Dirichlet boundary conditions in cuspidal domains. The investigation techniques are based on transforming the main problem, through a natural change of variables, into a complete abstract fourth-order differential equation involving fractional powers of linear operators, which allows us to provide results on well-posedness. Furthermore, we explore periodic-type solutions for fractional neutral evolution equations involving Caputo and -Hilfer derivatives, utilizing classical fixed point theorems as a preliminary step toward further investigation of fractional-order boundary value problems in non-smooth domains.