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

  • Hasib Khan
  • Hossein Jafari
  • Dumitru Baleanu
  • Rahmat Ali Khan
  • Aziz Khan
Original Research


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 \(\omega ,\epsilon \in \left( 3,4 \right] \). The coupled system is given by
$$\begin{aligned} D^{\omega }u\left( t \right)= & {} -G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\varepsilon }v\left( t \right)= & {} -G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\delta }u\left( 1 \right)= & {} 0=I^{3-\omega }u\left( 0 \right) =I^{4-\omega }u\left( 0 \right) , u(1)=\frac{{\Gamma }\left( {\omega -\delta } \right) }{{\Gamma }\left( \omega \right) }I^{\omega -\delta }\\&\quad G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) ,\\ D^{\nu }v\left( 1 \right)= & {} 0=I^{3-\varepsilon }v\left( 0 \right) =I^{4-\nu }v\left( 0 \right) , v(1)=\frac{{\Gamma }\left( {\varepsilon -\nu } \right) }{{\Gamma }\left( \varepsilon \right) }I^{\varepsilon -\nu }\\&\quad G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) , \end{aligned}$$
where \(t\in \left[ {0,1}\right] \), \(\delta ,\nu \in \left[ {1,2} \right] .\) The functions \(G_1 ,G_2 :\left[ {0,1} \right] \times R\times R\rightarrow R,\) satisfy the Caratheodory conditions. The fractional derivatives \(D^{\omega },D^{\varepsilon },D^{\delta },D^{\nu }\) are in Riemann-Liouville sense and \(I^{\omega },I^{\varepsilon },I^{3-\omega },I^{4-\omega },I^{3-\varepsilon },I^{4-\varepsilon },I^{\omega -\delta },I^{\varepsilon -\nu }\) 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.


Compliance with Ethical Standards

Conflict of interest

The authors have no competing interest regarding the publication of this article.


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Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Hydrology-Water Recourses and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and MaterialsHohai UniversityNanjingChina
  2. 2.Shaheed Benazir Bhutto University, SheringalSheringalPakistan
  3. 3.Department of MathematicsUniversity of MazandaranBabolsarIran
  4. 4.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa
  5. 5.Department of MathematicsCankaya UniversityBalgat, AnkaraTurkey
  6. 6.Institute of Space SciencesMagurele-BucharestRomania
  7. 7.University of Malaknd, ChakdaraChakdaraPakistan
  8. 8.Department of MathematicsUniversity of PeshawarPeshwarPakistan

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