Abstract
The purpose of this paper is to study the existence and uniqueness of the solution of nonlinear fractional differential equations with Mittag–Leffler nonsingular kernel. Two numerical methods to solve this problem are designed, and their stability and error estimates are investigated by discretizing the convolution integral and using the Grönwall’s inequality. Finally, the theoretical results are verified by using five illustrative examples.
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Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications. Academic Press, New York (1999)
Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Machado, J.A.T.: Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn. 62(1), 371–378 (2010)
Yang, X.J., Machado, J.A.T., Hristov, J.: Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow. Nonlinear Dyn. 84(1), 3–7 (2016)
Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: From the Cell to the Ecosystem. Wiley, New York (2014)
Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics, vol. 378. Springer, Wien (2014)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Models and Numerical Methods. World Scientific, Berlin (2012)
Abdelkawy, M.A., Zaky, M.A., Bhrawy, A.H., Baleanu, D.: Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model. Rom. Rep. Phys. 67(3), 773–791 (2015)
Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Academic Press, London (2015)
Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)
Liouville, J.: Mémoire sur quelques qustions de géomerie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions. J. d’École Polytechnique 1(3), 1–69 (1832)
Novikov, V., Wojciechowski, K., Komkova, O., Thiel T.: Anomalous relaxation in dielectrics. Equations with fractional derivatives. Mater. Sci. Pol. 23(4), 977–84 (2005)
Yang, X.J., Machado, J.A.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200–206 (2017)
Sibatov, R., Uchaikin, D.: Fractional relaxation and wave equations for dielectrics characterized by the Havriliak–Negami response function. In: Proceedings of the International Conference New Trends in Nanotechnology and Dynamical Systems, Turkey, Ankara, p. 15 (2010)
Garra, R., Gorenflo, R., Polito, F., Tomovski, Z.: Hilfer–Prabhakar derivatives and some applications. Appl. Math. Comput. 242, 576–89 (2014)
Kilbas, A., Saigo, M., Saxena, R.: Generalized Mittag–Leffler function and generalized fractional calculus operators. Integr. Transf. Spec. F. 15(1), 31–49 (2004)
Prabhakar, T.: A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)
Agila, A., Baleanu, D., Eid, R., Irfanoglu, B.: Applications of the extended fractional Euler–Lagrange equations model to freely oscillating dynamical systems. Rom. J. Phys. 61(3), 350–359 (2016)
Kumar, D., Singh, J., Baleanu, D.: A fractional model of convective radial fins with temperature-dependent thermal conductivity. Rom. Rep. Phys. 69, 103 (2017)
Yang, X.J.: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm. Sci. 21(3), 1161–1171 (2017)
Garrappa, R.: Grünwald-Letnikov operators for fractional relaxation in Havriliak–Negami models. Commun. Nonlinear Sci. Numer. Simul. 38, 178–191 (2016)
Yang, X.J., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20(2), 753–756 (2016)
Yang, X.J., Machado, J.A.T.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A 481, 276–283 (2017)
Yang, X.J., Gao, F., Machado, J.A.T., Baleanu, D.: A New Fractional Derivative Involving the Normalized Sinc Function Without Singular Kernel. arXiv:1701.05590 (2017)
Yang, X., Srivastava, H., Torres, D., Debbouche, A.: General fractional-order anomalous diffusion with non-singular power-law kernel. Therm. Sci. 21(1), S1–S9 (2017)
Gao, F.: General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems. Therm. Sci. 21(1), S11–S18 (2017)
Yang, X.J.: New general fractional-order rheological models with kernels of Mittag–Leffler functions. Rom. Rep. Phys. 69(4), 1–15 (2017)
Srivastava, H., Tomovski, Ž.: Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel. Appl. Math. Comput. 211, 198–210 (2009)
Tomovski, Ž., Hilfer, R., Srivastava, H.M.: Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions. Integr. Transf. Spec. F. 21(11), 797–814 (2010)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag–Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)
Miller, K.S., Samko, S.G.: A note on the complete monotonicity of the generalized Mittag–Leffler function. Real Anal. Exch. 23(2), 753–755 (1997–1998)
Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag–Leffler function and generalized fractional calculus operators. Integr. Transf. Spec. F. 15(1), 31–49 (2004)
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag–Leffler functions and their applications. J. Appl. Math. 2011 1–51 (2011), Article ID 298628
Pskhu, A.V.: On the theory of the continual integro-differentiation operator. Differ. Equ. 40(1), 128–136 (2004)
Pskhu, A.V.: Partial Differential Equations of Fractional Order. Nauka, Moscow (2005)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)
Ahokposi, D.P., Atangana, A., Vermeulen, D.P.: Modelling groundwater fractal flow with fractional differentiation via Mittag–Leffler law. Eur. Phys. J. Plus 132(165), 1–17 (2017)
Tateishi, A., Ribeiro, H., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. (2017). https://doi.org/10.3389/fphy.2017.00052
Baleanu, D., Fernandez, A.: On some new properties of fractional derivatives with Mittag–Leffler kernel. Commun. Nonlinear Sci. Numer. Simul. 59, 444–462 (2018)
Djida, J.D., Atangana, A., Area, I.: Numerical computation of a fractional derivative with non-local and non-singular kernel. Math. Model. Nat. Phenom. 12(3), 4–13 (2017)
Coronel-Escamilla, A., Aguilar, J., Dumitru, B., Escobar-Jimenez, R., Olivares-Peregrino, V., Abundez-Pliego, A.: Formulation of Euler–Lagrange and Hamilton equations involving fractional operators with regular kernel. Adv. Differ. Equ. 283, 1–17 (2016)
Abdeljawad, T., Baleanu, D.: Discrete fractional differences with nonsingular discrete Mittag–Leffler kernels. Adv. Differ. Equ. 232, 1–18 (2016)
Abdeljawad, T., Baleanu, D.: Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag–Leffler kernel. Chaos Soliton. Fract. 102, 106–110 (2017)
Wu, G.C., Baleanu, D., Zeng, S.D., Deng, Z.G.: Discrete fractional diffusion equation. Nonlinear Dyn. 80, 281–286 (2015)
Abdeljawad, T., Baleanu, D.: Integration by parts and its applications of a new nonlocal fractional derivative with Mittag–Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10, 1098–1107 (2017)
Li, C., Zeng, F.: The finite difference methods for fractional ordinary differential equations. Num. Funct. Anal. Opt. 34(2), 149–179 (2013)
Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)
Fei-Fei, J.: An equatorial ocean recharge paradigm for ENSO. Part I: conceptual model. J. Atmos. Sci. 54(7), 811–829 (1996)
Mo, J.Q., Lin, W.T., Zhu, J.: The variational iteration solving method for El Nino/La nino-southern oscillation model. Adv. Math. 35(2), 232–236 (2006)
Mo, J.Q., Lin, W.T.: Generalized variation iteration solution of an atmosphere-ocean oscillator model for global climate. J. Syst. Sci. Complex. 24(2), 271–276 (2011)
Singh, J., Kumar, D., Nieto, J.J.: Analysis of an El Nino-Southern Oscillation model with a new fractional derivative. Chaos Soliton. Fract. 99, 109–115 (2017)
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Baleanu, D., Jajarmi, A. & Hajipour, M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dyn 94, 397–414 (2018). https://doi.org/10.1007/s11071-018-4367-y
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DOI: https://doi.org/10.1007/s11071-018-4367-y