The New Exact and Approximate Solution for the Nonlinear Fractional Diffusive Predator-Prey system Arising in Mathematical Biology

Abstract

In this article two methods, q-Homotopy analysis Method (q-HAM) and Sine-Gordon expansion
method for solving Fractional Diffusive Predator-Prey system are proposed. The fractional derivative
is considered in the conformable sense. The solutions obtained using the suggested methods are in
very excellent agreement with the already existing ones and show that this approach can be solved the
problem effectively.

Keywords

• Sine-Gordon Expansion Method,Fractional Diffusive Predator-Prey system,q-Homotopy Analysis Method,Conformable Fractional Derivative

References

1. [1] Liao,S.J. The proposed homotopy analysis technique for the solution of nonlinear problems,Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
2. [2] El-Tawil, M. A. andHuseen, S.N. (2012)The q-Homotopy Analysis Method (qHAM), Inter-national Journal of Applied mathematics and mechanics, 8 (15): 51-75.
3. [3] El-Tawil, M. A. and Huseen, S.N., On Convergence of The q-Homotopy Analysis Method,Int. J. Contemp. Math. Sciences,8(10) 2013, 481 497.
4. [4] Iyiola O. S,(2013) q-Homotopy Analysis Method and Application to FingeroImbibitionphe-nomena in double phase ow through porous media, Asian Journal of CurrentEngineeringand Maths 2.283 - 286.
5. [5] Iyiola O. S., Soh M. E. and Enyi C. D., (2013)Generalized Homotopy Analysis Method(q-HAM) For Solving Foam Drainage Equation of Time Fractional Type, Mathematics inEngineering, Science and Aerospace (MESA), 4( 4),429- 440.
6. [6] Huseen S. N. and Grace S. R., Approximate Solutions of Nonlinear Partial DifferentialEquations by Modied q-Homotopy Analysis Method (mq-HAM), Hindawi Publishing Cor-poration, Journal of Applied Mathematics, (2013)Article ID 569674.
7. [7] Yan, C. A (1996). simple transformation for nonlinear waves. Physics Letters A. Volume224, 77-84.
8. [8] Zhen-Ya, Y., Hong-oing, Z., En-Gui, F.: New explicit and travelling wave solutions for aclass of nonlinear evolution equations. Acta Phys. Sin. 48(1)(1999) 15.

9. [9] Yel, G.. Baskonus, H. M., Bulut, H. (2017). Novel archetypes of new coupled KonnoOonoequation by using sineGordon expansion method, Opt Quant Electron 49:285.
10. [10] Baskonus, H.M., Sulaiman, T.A., Bulut, H.(2017)On the novel wave behaviors to the cou-pled nonlinear Maccaris system with complex structure. Opt. Int. J. Light Electron Opt.131, 10361043
11. [11] Zayed, E. M., Amer, Y. A. (2015). The modied simplequation method for solving nonlin-ear diffusive predator-prey system and Bogoyavlenskii equations. International Journal ofPhysical Sciences, 10(4), 133-141.
12. [12] Petrovskii, S., Malchow, H., Li, B. L. (2005, April). An exact solution of a diffusive predator-prey system. In Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences (Vol. 461, No. 2056, pp. 1029-1053). The Royal Society. Chicago.
13. [13] Kot, M. (2001). Elements of mathematical ecology. Cambridge University Press.
14. [14] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. (2014). A new definition of fractionalderivative, Journal of Computational and Applied Mathematics, 264, 65-70.
15. [15] Khodadad,F. S. Nazari,F. Eslami, M. Rezazadeh,H. (2017). Soliton solutions of the con-formable fractional Zakharov?Kuznetsov equation with dual-power law nonlinearity. Opticaland Quantum Electronics, 49(11), 384.
16. [16] Eslami, M. (2016). Exact traveling wave solutions to the fractional coupled nonlinearSchrodinger equations. Applied Mathematics and Computation, 285, 141-148.
17. [17] Hosseini, K., Mayeli, P., Ansari, R. (2017). Modified Kudryashov method for solving theconformable time-fractional Klein-Gordon equations with quadratic and cubic nonlineari-ties. Optik- International Journal for Light and Electron Optics, 130, 737-742.
18. [18] Khodadad, F. S., Nazari, F., Eslami, M., Rezazadeh, H. (2017). Soliton solutions of the con-formable fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity. Opticaland Quantum Electronics, 49(11), 384.
19. [19] Eslami, M., Khodadad, F. S., Nazari, F., Rezazadeh, H. (2017). The first integral methodapplied to the Bogoyavlenskii equations by means of conformable fractional derivative.Optical and Quantum Electronics, 49(12), 391.
20. [20] Aminikhah, H., Sheikhani, A. R., Rezazadeh, H. (2016). Sub-equation method for thefractional regularized long-wave equations with conformable fractional derivatives. Scienti-aIranica. Transaction B, Mechanical Engineering, 23(3), 1048.
21. [21] Rezazadeh, H., Ziabarya, B. P. (2016). Sub-equation method for the conformable fractionalgeneralized kuramotosivashinsky equation. Computational Research Progress in AppliedScience and Engineering, 2(3), 106-109.
22. [22] Rezazadeh, H.,Khodadad, F. S., Manafian, J. (2016). New structure for exact solutions ofnonlinear time fractional Sharma-Tasso-Olver equation via conformable fractional deriva-tive. Applications and Applied Mathematics: An International Journal, (accepted).
23. [23] Tariq, H., Akram, G. (2016). New traveling wave exact and approximate solutions for thenonlinear Cahn-Allen equation: evolution of a nonconserved quantity. Nonlinear Dynamics,1-14.
24. [24] Eslami, M., Rezazadeh, H. (2016). The first integral method for WuZhang system withconformable time-fractional derivative. Calcolo, 53(3), 475-485.
25. [25] Ekici, M., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S. P., Biswas, A., Belic,M. (2016). Optical soliton perturbation with fractional-temporal evolution by first integralmethod with conformable fractional derivatives. Optik-International Journal for Light andElectron Optics, 127(22), 10659-10669.
26. [26] Cenesiz, Y., Baleanu, D., Kurt, A., Tasbozan, O. (2016). New exact solutions of Burgers'type equations with conformable derivative. Waves in Random and Complex Media, 1-14.
27. [27] Kurt, A., Tasbozan, O., Cenesiz, Y.,Baleanu, D. (2016). New Exact Solutions for SomeNonlinear Conformable PDEs Using Exp-Function Method. In International Conference onApplied Mathematics and Analysis in Memory of Gusein Sh. Guseinov.Atilim UniversityAnkara, Turkey.
28. [28] Tasbozan, O., Cenesiz, Y., Kurt, A. (2016). New solutions for conformable fractional Boussi-nesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method.The European Physical Journal Plus, 131(7), 244.
29. [29] Eslami, M., Rezazadeh, H., Rezazadeh, M., Mosavi, S. S. (2017). Exact solutions tothe space-time fractional Schrodinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation. Optical and Quantum Electronics, 49(8), 279.
30. [30] Korkmaz, A., Hepson, O. E., Hosseinic, K., Rezazadeh, H., Eslami, M. (2017). On TheExact Solutions to Conformable Time Fractional Equations in EW Family Using Sine-Gordon Equation Approach.
31. [31] Hosseini, K., Bekir, A., Ansari, R. (2017). Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the exp(-φ("))-expantion method. Optical and Quan-tum Electronics, 49(4), 131.
32. [32] Korkmaz, A., Hepson, O. E., Hosseini, K., Rezazadehd, H., Eslami, M. (2017). Sine-GordonExpansion Method for Exact Solutions to Conformable Time Fractional Equations in RLW-Class.
33. [33] Cenesiz, Y., Tasbozan, O., Kurt, A. (2017). Functional Variable Method for conformablefractional modified KdV-ZK equation and Maccari system. Tbilisi Mathematical Journal,10(1), 117- 125.
34. [34] Kaplan, M., Bekir, A., Ozer, M. N. (2017). A simple technique for constructing exactsolutions to nonlinear differential equations with conformable fractional derivative. Opticaland Quantum Electronics, 49(8), 266.
35. [35] Hepson, O. E., Korkmaz, A., Hosseini, K., Rezazadeh, H., Eslami, M. (2017). An ExpansionBased on Sine-Gordon Equation to Solve KdV and modified KdV Equations in ConformableFractional Forms.
36. [36] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, (Chap-man and Hall/CRC Press, Boca Raton, 2003)
37. [37] Iyiola, O. S., Tasbozan, O., Kurt, A., Cenesiz, Y. (2017). On the analytical solutions ofthe system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos,Solitons and Fractals, 94, 1-7.
38. [38] El-Tawil, M. A., Huseen, S. N. (2012). The q-homotopy analysis method (q-HAM). Inter-national Journal of Applied mathematics and mechanics, 8(15), 51-75.