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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 |
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.
References:
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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. |
| [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. |
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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. |
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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. |
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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. |
| [6] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
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| [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. |
| [8] |
A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu,
On nonlinear fractional KleinGordon equation, Signal Processing, 91 (2011), 446-451.
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| [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. |
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J. Losada and J. J. Nieto,
Properties of a new fractional derivative without singular kernel, Progr Fract Differ Appl, 1 (2015), 87-92.
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| [11] |
I. Podlubny,
Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.
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| [12] |
B. Sambandham and A. Vatsala,
Basic results for sequential caputo fractional differential equations, Mathematics, 3 (2015), 76-91.
|
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
M. Caputo and M. Fabrizio,
A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
|
| [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. |
| [8] |
A. K. Golmankhaneh, A. K. Golmankhaneh and D. Baleanu,
On nonlinear fractional KleinGordon equation, Signal Processing, 91 (2011), 446-451.
|
| [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. |
| [10] |
J. Losada and J. J. Nieto,
Properties of a new fractional derivative without singular kernel, Progr Fract Differ Appl, 1 (2015), 87-92.
|
| [11] |
I. Podlubny,
Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367-386.
|
| [12] |
B. Sambandham and A. Vatsala,
Basic results for sequential caputo fractional differential equations, Mathematics, 3 (2015), 76-91.
|
| [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. |
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