Existence and uniqueness of solutions to fractional differential equations with fractional boundary conditions

Abstract

Boundary conditions involving fractional derivatives of unknown functions are more general and can be used to generalize Dirichlet- or Neumann-type boundary conditions. In this article, we consider a class of nonlinear fractional differential equations having Atangana-Baleanu derivatives equipped with fractional boundary conditions. We employ some standard fixed-point theorems to establish the main results — Leray–Schauder alternative ensures the existence of solutions, whereas Banach contraction principle guarantees uniqueness. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining a precise numerical estimation of the solution. Some examples are given to illustrate our analytical results and numerical findings. The main distinctive features of this work are as follows. (i) Some important properties of higher-order Atangana-Baleanu operators are introduced. (ii) Fractional boundary conditions are considered for the first time. (iii) A numerical approximation for the solution is expounded.