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:
[1]

A. A. M. ArafaS. 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-SayedA. ElsaidI. 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. GolmankhanehA. 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.

show all references

References:
[1]

A. A. M. ArafaS. 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-SayedA. ElsaidI. 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. GolmankhanehA. 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.

[1]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

[2]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713

[3]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

[4]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193

[5]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004

[6]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841

[7]

Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691

[8]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[9]

Franca Franchi, Barbara Lazzari, Roberta Nibbi. Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111

[10]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725

[11]

Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481

[12]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i

[13]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165

[14]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[15]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675

[16]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

[17]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239

[18]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

[19]

Sanjay Khattri. Another note on some quadrature based three-step iterative methods for non-linear equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 549-555. doi: 10.3934/naco.2013.3.549

[20]

Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541

2017 Impact Factor: 0.561

Article outline

[Back to Top]