# General solution of second order fractional differential equations

## DOI:

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

2018-05-20## Keywords:

Linear fractional differential equations, Conformable fractional derivative, Constant coefficients approach, Eulerâ€™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 Eulerâ€™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 â€“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. Ã‡enesiz, O. Tasbozan, On the Solution of Burgersâ€™ Equation with the New Fractional Derivative, Open Physics,13(1) (2015) 355-360.

[14] O.S. Iyiola, O. Tasbozan,, A. Kurt, Y. Ã‡enesiz, 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. Ã‡enesiz, O. Tasbozan, A. Kurt, Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system, Tbilisi Mathematical Journal, 10(1) (2017) 117â€“125.

[16] Y. Ã‡enesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgersâ€™ 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. Ã‡enesiz , 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 Blackâ€“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 SchrÃ¶dinger 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.

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