A numerical approach to the coupled atmospheric ocean model using a fractional operator


  • Pundikala Veeresha Center for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru-560029, India https://orcid.org/0000-0002-4468-3048




Caputo–Fabrizio derivative, El Nino-Southern oscillation model, fixed point theorem


In the present framework, the coupled mathematical model of the atmosphere-ocean system called El Nino-Southern Oscillation (ENSO) is analyzed with the aid Adams-Bashforth numerical scheme. The fundamental aim of the present work is to demonstrate the chaotic behaviour of the coupled fractional-order system. The existence and uniqueness are demonstrated within the frame of the fixed-point hypothesis with the Caputo--Fabrizio fractional operator. Moreover, we captured the chaotic behaviour for the attained results with diverse order. The effect of the perturbation parameter and others associated with the model is captured. The obtained results elucidate that, the present study helps to understand the importance of fractional order and also initial conditions for the nonlinear models to analyze and capture the corresponding consequence of the fractional-order dynamical systems.


Download data is not yet available.


M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, (1969).

K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, A Wiley, New York, (1993).

I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam,(2006).

D. Baleanu, Z.B. Guvenc, J.A. Tenreiro Machado, New trends in nanotechnology and fractional calculus applications, SpringerDordrecht Heidelberg, London New York, (2010).

M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Diff. Appl., 1 (2) (2015),73-85.

K. M. Safare, et.al., A mathematical analysis of ongoing outbreak COVID-19 in India through nonsingular derivative, Numerical Methods for Partial Differential Equations 37 (2) (2021), 1282-1298.

M. Yavuz, European option pricing models described by fractional operators with classical and generalized Mittag-Lefflerkernels, Numerical Methods for Partial Differential Equations, (2021), DOI: 10.1002/num.22645.

L. Akinyemi, M. Şenol, S. N. Huseen, Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsovequation in dusty plasma, Adv. Differ. Equ., 45 (2021), DOI: 10.1186/s13662-020-03208-5.

C. Baishya, S. J. Achar, P. Veeresha, D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection both the populations, Chaos, 31 (2021), DOI: 10.1063/5.0028905.

J. Fei-Fei, An equatorial ocean recharge paradigm for ENSO. Part I: conceptual model, J. Atmos. Sci., 54 (7) (1996), 811-829.

Y. Zen, The Laplace-Adomian-Pade technique for the ENSO model, Math. Probl. Eng, 4 (2013), DOI:10.1155/2013/954857.

J. Q. Mo, W. T. Lin, J. Zhu, The variational iteration solving method for El Nino/La Nino-Southern Oscillation model, Adv.Math., 35 (2) (2006), 232–236.

J. Q. Mo, W. T. Lin, Generalized variation iteration solution of an atmosphere-ocean oscillator model for global climate, J.Syst. Sci. Complex, 24 (2) (2011), 271-276.

Z. Xian-Chun, L. Yi-Hua, W. T. Lin, J. Q. Mo, Homotopic mapping solution of an oscillator for the El nino/La Nina-SouthernOscillation, Chin. Phys. B, 18 (11) (2009), 4603-4605.

J. Singh, D. Kumar, J. J. Nieto, Analysis of an El Nino-Southern Oscillation model with a new fractional derivative, ChaosSolitons Fractals, 99 (2017), 109-115.

M. Gubes, H. A. Peker, G. Oturanc, Application of differential transform method for El Nino Southern Oscillation (ENSO)model with compared Adomian decomposition and variational iteration methods. J. Math. Comput. Sci., 15 (2015), 167–178.

J. Q. Mo JQ, W. T. Lin, Perturbed solution for the ENSO nonlinear model, Acta Phys. Sinica., 53 (4) (2004), 996-998.

L. Akinyemi, O.S. Iyiola, A reliable technique to study nonlinear time-fractional coupled Korteweg-de Vries equations, Adv.Differ. Equ., 2020 (2020), 1-27, DOI: 10.1186/s13662-020-02625-w.

E. K. Akgül, A. Akgül, M. Yavuz, New Illustrative Applications of Integral Transforms to Financial Models with DifferentFractional Derivatives, Chaos Solitons Fractals 146 (2021), 110877.

P. Veeresha, E. Ilhan, H. M. Baskonus, Fractional approach for analysis of the model describing wind- influenced projectile motion, Phys. Scr., 96 (2021), DOI: 10.1088/1402-4896/abf868.

L. Akinyemi, A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction, Comp. Appl. Math.,39 (2020), 1-34, DOI: 10.1007/s40314-020-01212-9.

P. Veeresha, D. G. Prakasha, A reliable analytical technique for fractional Caudrey-Dodd-Gibbon equation with Mittag-Lefflerkernel, Nonlinear Eng., 9 (1) (2020), 319–328.

M. Yavuz, N. Sene, Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, J. Ocean Eng. Sci., 6 (2) (2021), 196-205.

S.-W. Yao, E. Ilhan, P. Veeresha, H. M. Baskonus, A powerful iterative approach for quintic complex Ginzburg-Landau equation within the frame of fractional operator, Fractals, (2021), DOI: 10.1142/S0218348X21400235.

L. Akinyemi, P. Veeresha, M. Senol, Numerical solutions for coupled nonlinear Schrodinger-Korteweg-de Vries and Maccari’ssystems of equations, Modern Physics Letters B, (2021), 2150339, DOI: 10.1142/S0217984921503395.

A. Atangana, R. T. Alqahtani, Numerical approximation of the space-time Caputo–Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Diff. Equ., 1 (2016), 1–13.

C. Baishya, Dynamics of a Fractional Stage Structured Predator-Prey model with Prey Refuge, Indian J. Ecol., 47 (4) (2020),1118-1124.

P. Veeresha, D.G. Prakasha, H.M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos 29 (013119) (2019). DOI: 10.1063/1.5074099.

K. M. Owolabi, A. Atangana, Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizioderivative, Chaos, Solitons Fractals, 105 (2017), 111–119.

A. Atangana, Derivative with a new parameter: theory, methods and applications, New York: Academic Press; 2016.

A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13(2018), DOI: 10.1051/mmnp/2018010.

A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7(2015), 1–6.

M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract.Diff. Appl., 2 (2016), 1–11.

J. Losada, J. J. Nieto, Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92.

J. Danane, K. Allali, Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Solitons Fractals, 136 (2020).

K. M. Owolabi, Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative, Phys. A, 523 (2019), 1072-1090.

F. Haq, I. Mahariq, T. Abdeljawad, N. Maliki, A new approach for the qualitative study of vector born disease using Caputo–Fabrizio derivative, Numer. Methods Partial Differ. Equ., 37 (2) (2021), 1809-1818.

P. Veeresha, H. M. Baskonus, W. Gao, Strong interacting internal waves in rotating ocean: Novel fractional approach, Axioms,10 (2) (2021).

W. Zhong, L. Wang, T. Abdeljawad, Separation and stability of solutions to nonlinear systems involving Caputo–Fabrizioderivatives, Adv. Differ. Equ., 166 (2020). DOI: 10.1186/s13662-020-02632-x.

R. Gul, M. Sarwar, K. Shah, T. Abdeljawad, F. Jarad, Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations, J. Funct. Spaces, (2020), DOI: 10.1155/2020/4714032.

M. Yavuz, E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.

P. Veeresha, D. G. Prakasha, Z. Hammouch, An efficient approach for the model of thrombin receptor activation mechanism with Mittag-Leffler function, Nonlinear Analysis: Problems, Applications and Computational Methods, (2020), 44-60.

K. Shah, F. Jarad, T. Abdeljawad, On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizioderivative, Alexandria Eng. J., 59 (4) (2020), 2305-2313.

M. Yavuz, N. Sene, Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, J. Ocean Eng. Sci., 6 (2) (2021), 196-205.




How to Cite

Veeresha, P. (2021). A numerical approach to the coupled atmospheric ocean model using a fractional operator. Mathematical Modelling and Numerical Simulation With Applications, 1(1), 1–10. https://doi.org/10.53391/mmnsa.2021.01.001



Research Articles