# Construction of new exact solutions to time-fractional two-component evolutionary system of order 2 via different methods

Article

## Abstract

This paper is concerned with the applications of five different methods including the sub-equation method, the tanh method, the modified Kudryashov method, the $$\left( \frac{G'}{G}\right)$$-expansion method and the Exp-function method to construct exact solutions of time-fractional two-component evolutionary system of order 2. We first convert this type of fractional equations to the nonlinear ordinary differential equations by means of fractional complex transform. Then, the five methods are adopted to solve the nonlinear ordinary differential equations. As a result, some new exact solutions are obtained. It is also shown that each of the considered methods can be used as an alternative for solving fractional differential equations.

## Keywords

Exact solutions Modified Riemann–Liouville derivative The time-fractional two-component evolutionary system of order 2

## Notes

### Acknowledgements

This work is supported by National Natural Science Foundation of China (No. 11601192), Natural Science Foundation of Jiangsu Province (No. BK20140522), Scientific Research Fund of Jiangsu University of Science and Technology.

## References

1. Ablowitz, M.J., Clarkson, P.A.: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)
2. Alquran, M.: Solitons and periodic solutions to nonlinear partial differential equations by the sine–cosine method. Appl. Math. Inf. Sci. 6(1), 85–88 (2012)
3. Alquran, M.: Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J Appl. Anal. Comput. 5(4), 589–599 (2015)
4. Alquran, M., Al-Khaled, K., Ananbeh, H.: New soliton solutions for systems of nonlinear evolution equations by the rational sine–cosine method. Stud. Math. Sci. 3(1), 1–9 (2011)Google Scholar
5. Bekir, A., Aksoy, E., Cevikel, A.C.: Exact solutions of nonlinear time fractional partial differential equations by sub-equation method. Math. Method Appl. Sci. 38(13), 2779–2784 (2015)
6. Bhrawy, A., Zaky, M.: An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)
7. Ege, S.M., Misirli, E.: The modified Kudryashov method for solving some fractional-order nonlinear equations. Adv. Differ. Equ. 2014(1), 135 (2014)
8. Eslami, M., Neirameh, A.: New exact solutions for higher order nonlinear Schrödinger equation in optical fibers. Opt. Quantum Electron. 50(1), 47 (2018)
9. Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88(2), 177–184 (2014)
10. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)
11. Foursov, M.V.: Classification of certain integrable coupled potential KdV and modified KdV-type equations. J. Math. Phys. 41(9), 6173–6185 (2000)
12. Foursov, M.V., Maza, M.M.: On computer-assisted classification of coupled integrable equations. J. Symb. Comput. 33, 647–660 (2002)
13. Guner, O., Aksoy, E., Bekir, A., Cevikel, A.C.: Different methods for $$(3+1)$$-dimensional space-time fractional modified Kdv–Zakharov–Kuznetsov equation. Comput. Math. Appl. 71(6), 1259–1269 (2016)
14. Guner, O., Atik, H., Kayyrzhanovich, A.A.: New exact solution for space-time fractional differential equations via $$(G^{\prime }/G)$$-expansion method. Optik 130, 696–701 (2017)
15. He, J.H., Elagan, S., Li, Z.: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376(4), 257–259 (2012)
16. Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51(9–10), 1367–1376 (2006)
17. Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22(3), 378–385 (2009)
18. Korkmaz, A., Hepson, O.E.: Traveling waves in rational expressions of exponential functions to the conformable time fractional Jimbo–Miwa and Zakharov–Kuznetsov equations. Opt. Quantum Electron. 50(1), 42 (2018)
19. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)
20. Lopes, A.M., Machado, J.T., Pinto, C.M., Galhano, A.M.: Fractional dynamics and MDS visualization of earthquake phenomena. Comput. Math. Appl. 66(5), 647–658 (2013)
21. Lu, D., Seadawy, A.R., Khater, M.M.: Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods. Results Phys. 7, 2028–2035 (2017)
22. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, NY (1993)
23. Obeidat, N.A., Rawashdeh, M.S., Alquran, M.: An improved approximate solutions to nonlinear partial differential equations using differential transform method and adomian decomposition method. Thai J. Math. 12(3), 569–589 (2014)
24. Perdikaris, P., Karniadakis, E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42(5), 1012–1023 (2014)
25. Sahoo, S., Ray, S.S.: A new method for exact solutions of variant types of time-fractional Korteweg–de Vries equations in shallow water waves. Math. Methods Appl. Sci. 40(1), 106–114 (2017)
26. Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Physica A 473, 352–362 (2017)
27. Wang, L., Wang, F.: Approximate solutions for time-fractional two-component evolutionary system of order 2 using coupled fractional reduced differential transform method. J. Appl. Anal. Comput. 7(4), 1312–1322 (2017)
28. Wang, M., Li, X., Zhang, J.: The $$(G^{\prime }/G)$$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)
29. Wang, W., Chen, X., Ding, D., Lei, S.L.: Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int. J. Comput. Math. 92(12), 2596–2614 (2015)
30. Yin, C., Cheng, Y., Zhong, S.M., Bai, Z.: Fractional-order switching type control law design for adaptive sliding mode technique of 3D fractional-order nonlinear systems. Complexity 21(6), 363–373 (2016)

## Authors and Affiliations

• Linjun Wang
• 1
• Wei Shen
• 1
• Yiping Meng
• 2
• Xumei Chen
• 1
1. 1.Faculty of ScienceJiangsu UniversityZhenjiangChina
2. 2.School of ScienceJiangsu University of Science and TechnologyZhenjiangChina