Study of Certain Fractional Differential Equations

Abstract

In this doctoral dissertation, we investigate the existence, uniqueness, and stability of solutions for various classes of nonlinear initial and boundary value problems (Pantograph, Langevin, Logistic) involving the variable order fractional operators. All conclusions drawn in the present research have been proven utilizing the variable order fractional calculus and fixed point theorem using the piecewise constant functions properties, which are crucial to convert the considered problems into an equivalent standard constant order counterparts. Furthermore, we investigate the stability in terms of Ulam-Hyers-Rassias stability criterion, and under further assumptions on the nonlinear term, we obtain the generalized Lyapunov inequalities.