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

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

Ilknur Koca

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.

Keywords: Non-linear equations, existence and uniqueness, Atangana-Baleanu fractional differentiation, numerical analysis.
Mathematics Subject Classification: Primary: 34A34, 34A12, 34A08; Secondary: 47N40.
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:
[1]

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

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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

[3]

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

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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

[5]

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

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M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[7]

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

[8]

A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu, On nonlinear fractional KleinGordon equation, Signal Processing, 91 (2011), 446-451.   Google Scholar

[9]

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

[10]

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

[11]

I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.   Google Scholar

[12]

B. Sambandham and A. Vatsala, Basic results for sequential caputo fractional differential equations, Mathematics, 3 (2015), 76-91.   Google Scholar

[13]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[7]

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

[8]

A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu, On nonlinear fractional KleinGordon equation, Signal Processing, 91 (2011), 446-451.   Google Scholar

[9]

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

[10]

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

[11]

I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.   Google Scholar

[12]

B. Sambandham and A. Vatsala, Basic results for sequential caputo fractional differential equations, Mathematics, 3 (2015), 76-91.   Google Scholar

[13]

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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