General solution of second order fractional differential equations

Authors

  • Mousa Ilie Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
  • Jafar Biazar Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O.Box.41335-1914, Guilan, Rasht, Iran
  • Zainab Ayati Department of Engineering sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran

DOI:

https://doi.org/10.14419/ijamr.v7i2.10116

Published:

2018-05-20

Keywords:

Linear fractional differential equations, Conformable fractional derivative, Constant coefficients approach, Euler’s equation, Variation of parameters, Lagrange method, Undetermined coefficients,

Abstract

Fractional differential equations are often seeming perplexing to solve. Therefore, finding comprehensive methods for solving them sounds of high importance. In this paper, a general method for solving second order fractional differential equations has been presented based on conformable fractional derivative. This method realizes on determining a general solution of homogeneous and a particular solution of a second order linear fractional differential equations. Furthermore, a general solution has been developed for fractional Euler’s equation. For more explanation of each part, some examples have been solved. 

References

[1] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65-70.

[2] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015) 57-66.

[3] V. Daftardar –Gejji, H. Jafari, Solving a multi- order fractional differential equation using Adomian Decomposition, Applied Mathematics and Computation, 189 (2007) 541-548.

[4] B. Ghazanfari, A. Sepahvandzadeh, Adomian Decomposition Method for solving Fractional Bratu-type equations, Journal of mathematics and computer science, 8 (2014) 236-244.

[5] O. Abdulaziz, I. Hashim, S. Momani, solving systems of fractional differential equations by homotopy perturbation method, Physics Letters 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, Journal of mathematics and computer science, 3 (2011) 212-224.

[7] M. Mahmoudi, M. V. Kazemi, Solving singular BVPs Ordinary Differential Equations by Modified Homotopy Perturbation Method, Journal of mathematics and computer science, 7 (2013) 138-143.

[8] M. Rabbani, New Homotopy Perturbation Method to Solve Non-Linear Problems, Journal of mathematics and computer science, 7 (2013) 272-275.

[9] I. Hashim, O. Abdulaziz, S. Momani, Homotopy Analysis Method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 674-684.

[10] A. Kurt, O. Tasbozan,Y. Cenesiz, Homotopy Analysis Method for Conformable Burgers-Korteweg-de Vries Equation, Bulletin of Mathematical Sciences and Applications, 17 (2016) 17-23.

[11] G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Physics Letters A, 374 (2010) 2506-2509.

[12] Z. Odibat, S. Momani, V. Suat Erturk, Generalized differential transform method: application to differential equations of fractional order, Applied Mathematics and Computation, 197 (2008) 467-477.

[13] A. Kurt, Y. Çenesiz, O. Tasbozan, On the Solution of Burgers’ Equation with the New Fractional Derivative, Open Physics,13(1) (2015) 355-360.

[14] O.S. Iyiola, O. Tasbozan,, A. Kurt, Y. Çenesiz, On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion, Chaos, Solitons & Fractals, 94 (2017) 1-7.

[15] Y. Çenesiz, O. Tasbozan, A. Kurt, Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system, Tbilisi Mathematical Journal, 10(1) (2017) 117–125.

[16] Y. Çenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27(1) (2017) 103-116.

[17] M. Ilie, J. Biazar, Z. Ayati, General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative, International Journal of Applied Mathematical Research, 6(2) (2017) 49-51.

[18] M. Ilie, J. Biazar, Z. Ayati, Application of the Lie Symmetry Analysis for second-order fractional differential equations, Iranian Journal of Optimization, 9(2) (2017) 79-83.

[19] Y. Çenesiz , A. Kurt, New fractional complex transform for conformable fractional partial differential equations, Journal of Applied Mathematics Statistics and Informatics, 12(2) (2016) 41-47.

[20] M. N. Koleva, L. G. Vulkov, Numerical solution of time-fractional Black–Scholes equation, Computational and Applied Mathematics, 39 (4) (2017) 1699-1715.[21] B. P. Moghaddam, M. J. A. T. Machado, A. Babaei, A computationally efficient method for tempered fractional differential equations with application, Computational and Applied Mathematics, 18 Nov 2017 online, 1-15.

[22] A. Gokdogan, E. Unal, E. Celik, Existence and uniqueness theorems for sequential linear conformable fractional equations, Miskolc Mathematical Notes, 17(1) (2016) 267-279.

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

[24] M. Ilie, J. Biazar, Z. Ayati, Analytical solutions for conformable fractional Bratu-type equations, International Journal of Applied Mathematical Research, 7 (1) (2018) 15-19.

[25] M. Ilie, J. Biazar, Z. Ayati, Optimal Homotopy Asymptotic Method for first-order conformable fractional differential equations, Journal of Fractional Calculus and Applications, 10 (1) (2019) 33-45.

[26] M. Ilie, J. Biazar, Z. Ayati, The first integral method for solving some conformable fractional differential equations, Optical and Quantum Electronics, 50 (2) (2018), https://doi.org/10.1007/s11082-017-1307-x.

[27] M. Ilie, J. Biazar, Z. Ayati, Resonant solitons to the nonlinear Schrödinger equation with different forms of nonlinearities, Optik, 164 (2018) 201-209.

[28] M. Ilie, J. Biazar, Z. Ayati, Lie Symmetry Analysis for the solution of first-order linear and nonlinear fractional differential equations, International Journal of Applied Mathematical Research, 7 (2) (2018) 37-41.

[29] M. Ilie, J. Biazar, Z. Ayati, Analytic solution for second-order fractional differential equations via OHAM, Journal of Fractional Calculus and Applications, 10 (1) (2019) 105-119.

View Full Article:

License

Authors who publish with this journal agree to the following terms:

                           [1]            Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

                           [2]            Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

                           [3]            Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).