Quasilinearization and boundary value problems for Riemann-Liouville fractional differential equations

Abstract

We apply the quasilinearization method to a Dirichlet boundary value problem and to a right focal boundary value problem for a Riemann-Liouville fractional differential equation. First, we sue the method of upper and lower solutions to obtain the uniqueness of solutions of the Dirichlet boundary value problem. Next, we apply a suitable fixed point theorem to establish the existence of solutions. We develop a quasilinearization algorithm and construct sequences of approximate solutions that converge monotonically and quadratically to the unique solution of the boundary value problem. Two examples are exhibited to illustrate the main result for the Dirichlet boundary value problem.

Keywords

• Riemann-Liouville fractional differential equation
• Dirichlet boundary value problem
• right focal boundary value problem
• upper and lower solutions, quasilinearization