On Iterative Solutions and Error Estimations of a Coupled System of Fractional Order Differential-Integral Equations with Initial and Boundary Conditions

Abstract

The study of boundary value problems (BVPs) for fractional differential-integral equations (FDIEs) is extremely popular in the scientific community. Scientists are utilizing BVPs for FDIEs in day life problems by the help of different approaches. In this paper, we apply monotone iterative technique for the existence, uniqueness and the error estimations of solutions for a coupled system of BVPs for FDIEs of orders omega, epsilon. (3, 4]. The coupled system is given by D(omega)u (t) = -G(1) (t, I(omega)u (t), I(epsilon)v (t)), D-epsilon v (t) = -G(2) (t, I(omega)u (t), I-epsilon v (t)), D(delta)u (1) = 0 = I(3-omega)u (0) = I(4-omega)u (0), u(1) = Gamma(omega - d) /Gamma(omega) I omega-delta G(1)(t, I(omega)u (t), I(epsilon)v(t)) (t = 1), D(nu)v (1) = 0 = I3-epsilon v (0) = I4-nu v (0), v(1) = Gamma(epsilon - nu)/Gamma(epsilon) I epsilon-nu G(2)(t, I(omega)u (t), I-epsilon v (t)) (t = 1), where t is an element of [0, 1], delta, nu is an element of [1, 2]. The functions G(1), G(2) : [0, 1] x R x R. R, satisfy the Caratheodory conditions. The fractional derivatives D-omega, D-epsilon, D-delta, D-nu are in Riemann-Liouville sense and I-omega, I-epsilon, I3-omega, I4-epsilon, I3-epsilon, I4-epsilon, I omega-delta, I epsilon-nu are fractional order integrals. The assumed technique is a better approach for the existence, uniqueness and error estimation. The applications of the results are examined by the help of examples.