General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative

Authors

  • Mousa Ilie

    Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
  • Jafar Biazar

    Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan
  • Zainab Ayati

    Department of Engineering sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran

Received date: November 25, 2016

Accepted date: December 19, 2016

Published date: April 29, 2017

DOI:

https://doi.org/10.14419/ijamr.v6i2.7014

Keywords:

Fractional Differential Equations, Conformable Fractional Derivative, Bernoulli Fractional Differential Equation, Riccati Fractional Differential Equation.

Abstract

This paper aimed to develop two well-known nonlinear ordinary different equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.

References

  1. [1] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014). https://doi.org/10.1016/j.cam.2014.01.002.

    [2] T. Abdeljawad, on conformable fractional calculus, Journal of Computational and Applied Mathematics., 279, 57-66 (2015). https://doi.org/10.1016/j.cam.2014.10.016.

    [3] V. Daftardar –Gejji, H. Jafari, Solving a multi- order fractional differential equation using Adomian Decomposition, Appl. Math. Comput., 189 (2007) 541-548. https://doi.org/10.1016/j.amc.2006.11.129.

    [4] B. Ghazanfari, A. Sepahvandzadeh, Adomian Decomposition Method for solving Fractional Bratu-type equations, J. Math. Computer and Science, 8 (2014) 236-244.

    [5] O. Abdulaziz, I. Hashim, S. Momani, solving systems of fractional differential equations by homotopy perturbation method, Phys. Let., A 372 (2008) 451-459.

    [6] B. Ghazanfari, A. G. Ghazanfari, M. Fuladvand, Modification of the Homotopy Perturbation Method for numerical solution of Nonlinear Wave and of Nonlinear Wave Equations., J. Math. Computer Sci. 3 (2011) 212-224.

    [7] M. Rabbani, New Homotopy Perturbation Method to Solve Non-Linear Problems, J. Math. Computer Sci., 7 (20013) 272-275.

    [8] I. Hashim, O. Abdulaziz, S. Momani, Homotopy Analysis Method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 674-684. https://doi.org/10.1016/j.cnsns.2007.09.014.

    [9] G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett., A374 (2010) 2506-2509. https://doi.org/10.1016/j.physleta.2010.04.034.

    [10] Z. Odibat, S. Momani, V. Suat Erturk, Generalized differential transform method: application to differential equations of fractional order, Appl. Math. Comput., 197 (2008) 467-477. https://doi.org/10.1016/j.amc.2007.07.068.

    [11] Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput, 215 (2009) 524-529. https://doi.org/10.1016/j.amc.2009.05.018.

    [12] Hossein. Aminikhah, Amir Hossein Refahi Sheikhani, Hadi Rezazadeh. Approximate analytical solution of distributed order fractional Riccati differential equation, Ain Shams Eng J (2016).

    [13] H. Aminikhah, A. H. Refahi sheikhani, H. Rezazadeh, Sub-equation method for the fractional regularized long-wave equations whit conformable fractional derivatives, Scientia Iranica B (2016) 23(3), 1048-1054.

    [14] Bothayna S.H. Kashkari, Muhammed I. Syam, Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order, Applied Mathematics and Computation 290 (2016) 281–291 https://doi.org/10.1016/j.amc.2016.06.003.

    [15] George F. Simmons, Differential Equations Whit Applications And Historical Notes, McGraw-Hill,Inc. New York. 1974.

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How to Cite

Ilie, M., Biazar, J., & Ayati, Z. (2017). General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative. International Journal of Applied Mathematical Research, 6(2), 49-51. https://doi.org/10.14419/ijamr.v6i2.7014

Received date: November 25, 2016

Accepted date: December 19, 2016

Published date: April 29, 2017