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 ω, ϵ∈ (3 , 4]. The coupled system is given by Dωu(t)=-G1(t,Iωu(t),Iεv(t)),Dεv(t)=-G2(t,Iωu(t),Iεv(t)),Dδu(1)=0=I3-ωu(0)=I4-ωu(0),u(1)=Γ(ω-δ)Γ(ω)Iω-δG1(t,Iωu(t),Iεv(t))(t=1),Dνv(1)=0=I3-εv(0)=I4-νv(0),v(1)=Γ(ε-ν)Γ(ε)Iε-νG2(t,Iωu(t),Iεv(t))(t=1),where t∈ [0 , 1] , δ, ν∈ [1 , 2]. The functions G1, G2: [0 , 1] × R× R→ R, satisfy the Caratheodory conditions. The fractional derivatives Dω, Dε, Dδ, Dν are in Riemann-Liouville sense and Iω, Iε, I3-ω, I4-ω, I3-ε, I4-ε, Iω-δ, Iε-ν 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. © 2017, Foundation for Scientific Research and Technological Innovation.