Two-point nonlocal nonlinear fractional boundary value problem with Caputo derivative: Analysis and numerical solution

Abstract This work presents the existential and unique results for the solution to a kind of high-order fractional nonlinear differential equations involving Caputo fractional derivative. The boundary condition is of the integral type, which entangles both starting and ending points of the domain. First, the unique exact solution is extracted in terms of Green’s function for the linear fractional differential equation, and then Banach contraction mapping theorem is applied to prove the main result in the case of the general nonlinear source term. Then, our main result is demonstrated by an illustrative example, which shows its legitimacy and applicability. Furthermore, numerical-based semi-analytical technique has been presented to approximate the unique solution to the desired order of precision.