On Iterative Solutions and Error Estimations of a Coupled System of Fractional Order Differential-Integral Equations wit
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On Iterative Solutions and Error Estimations of a Coupled System of Fractional Order Differential-Integral Equations with Initial and Boundary Conditions Hasib Khan 1,2 · Hossein Jafari 3,4 · Dumitru Baleanu4,5,6 · Rahmat Ali Khan 7 · Aziz Khan 8
© Foundation for Scientific Research and Technological Innovation 2017
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, ∈ (3, 4]. The coupled system is given by D ω u (t) = −G 1 t, I ω u (t) , I ε v (t) , D ε v (t) = −G 2 t, I ω u (t) , I ε v (t) , D δ u (1) = 0 = I 3−ω u (0) = I 4−ω u (0) , u(1) = G 1 t, I ω u (t) , I ε v (t) (t = 1) ,
B
(ω − δ) ω−δ I (ω)
Hossein Jafari [email protected]
1
State Key Laboratory of Hydrology-Water Recourses and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 210098, Jiangsu, China
2
Shaheed Benazir Bhutto University, Sheringal, Dir Upper, P. O. Box 18000, Sheringal, Khyber Pakhtunkhwa, Pakistan
3
Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
4
Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa
5
Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey
6
Institute of Space Sciences, P.O. BOX, MG-23, 76900 Magurele-Bucharest, Romania
7
University of Malaknd, Chakdara, Dir Lower, P. O. Box 18000, Chakdara, Khyber Pakhtunkhwa, Pakistan
8
Department of Mathematics, University of Peshawar, Peshwar, Khyber Pakhtunkhwa, Pakistan
123
Differ Equ Dyn Syst
D ν v (1) = 0 = I 3−ε v (0) = I 4−ν v (0) , v(1) = G 2 t, I ω u (t) , I ε v (t) (t = 1) ,
(ε − ν) ε−ν I (ε)
where t ∈ [0, 1], δ, ν ∈ [1, 2] . The functions G 1 , G 2 : [0, 1] × R × R → R, satisfy the Caratheodory conditions. The fractional derivatives D ω , D ε , D δ , D ν are in Riemann-Liouville sense and I ω , I ε , I 3−ω , I 4−ω , I 3−ε , I 4−ε , I ω−δ , I ε−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.
Introduction Fractional differential equations (FDEs) are generalization of the differential equations of integer orders. This generalization has attracted the attention of scientists in engineering and science in different disciplines including polymer chemistry, biophysics, aerodynamics, control theory, mechanics etc [1–3]. Different areas of FDEs are under consideration for research work and from the available literature one can observe the attention of researchers in the existence theory by the
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