Study of Nonlocal Differential Equations

Abstract

This thesis examines the theory, classification, and solution behavior of second-, third, and fourth-order nonlocal differential equations. Nonlocal differential equations, which are distinguished by the occurrence of terms involving integration or dependence on values of the unknown function over an interval or the whole domain, have attracted considerable attention because of their applications in mathematical physics, control theory, and continuum mechanics. We starts with a thorough examination of second-order nonlocal linear differential equations, their overall form, solution techniques, and stability behavior. This is then generalized to third-order nonlocal equations, where complexity increases through derivatives of order greater than one and increased functional dependence. Lastly, a complete framework is established for fourth-order nonlocal differential equations, giving homogeneous and nonhomogeneous cases with different nonlocal kernels. Every order is analyzed with example cases motivated by current literature, and closed-form solutions when possible are obtained. Existence, uniqueness, and qualitative solution behavior are also considered. The thesis adds to the general comprehension of higher-order nonlocal operators and gives one a single way of solving and analyzing these equations systematically