    June  2019, 12(3): 475-486. doi: 10.3934/dcdss.2019031

## Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative

 Mehmet Akif Ersoy University, Department of Mathematics, Faculty of Sciences, 15100, Burdur, Turkey

Received  June 2017 Revised  September 2017 Published  September 2018

A nonlinear system of two fractional nonlinear differential equations with Atangana-Baleanu derivative is considered in this work. General conditions under which a system solution exists and unique are presented using the fixed-point theorem method. The well-established numerical scheme is used to solve the system of equations. A numerical analysis is presented to secure the stability and convergence of the used numerical scheme.

Citation: Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031
##### References:
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##### References:
  A. A. M. Arafa, S. Z. Rida and H. Mohamed, Homotopy analysis method for solving biological population model, Communications in Theoretical Physics, 56 (2011), 797-800.  doi: 10.1088/0253-6102/56/5/01.  Google Scholar  A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. Google Scholar  A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar  A. Atangana and I. Koca, On the new fractional derivative and application to Nonlinear Baggs and Freedman model, Journal of Nonlinear Sciences and Applications, 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46.  Google Scholar  A. Atangana, On the new fractional derivative and application to nonlinear fisher's reaction-diffusion equation, Appl Math Comput, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar  M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar  A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla and D. Hammad, A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Applied Mathematics and Computation, 218 (2012), 8329-8340.  doi: 10.1016/j.amc.2012.01.057.  Google Scholar  A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu, On nonlinear fractional KleinGordon equation, Signal Processing, 91 (2011), 446-451.   Google Scholar  A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006. Google Scholar  J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr Fract Differ Appl, 1 (2015), 87-92.   Google Scholar  I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.   Google Scholar  B. Sambandham and A. Vatsala, Basic results for sequential caputo fractional differential equations, Mathematics, 3 (2015), 76-91.   Google Scholar  T. Yamamoto and X. Chen, An existence and nonexistence theorem for solutions of nonlinear systems and its application to algebraic equations, Journal of Computational and Applied Mathematics, 30 (1990), 87-97.  doi: 10.1016/0377-0427(90)90008-N.  Google Scholar
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