The modified simple equation method for the two space-time nonlinear fractional partial differential equations

Document Type : Research Paper

Authors

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

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan Rasht,Iran

3 Department of Mathematics, Darab Branch, Islamic Azad University Darab,Iran

Abstract

Many important phenomena in various fields are described and generalized by a fractional partial differential equation.
In this paper, the modified simple equation method which is widely applicable to handle nonlinear wave equations, is successfully implemented for constructing exact solutions of two nonlinear fractional equations, namely the space-time nonlinear fractional potential Kadomstev- Petviashvili (PKP) and Sharma-Tasso- Olver (STO) equations in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions are successfully obtained for them

Keywords

References

1] J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some fractional PDEs , Amer. J. Math. Anal. 1 (2013) 14-19.
[2] I. Aslan, The  rst integral method for constructing exact and explicit solutions to nonlinear evolution equations , Math. Methods Appl. Sci., 35 (2012) 716722.
[3] I. Aslan, Some exact and eplicit solutions for nonlinear Schrdinger equations, Acta Phys. Pol. A, 123 (1) (2013) 1620.
[4] K. Assaleh, W.M. Ahmad, Modeling of speech signals using fractional calculus , 9th International Symposium on Signal Processing and Its Applications, 12(15) (2007) 1- 4.
[5] A. Bekir, O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana. J. Phys. 79 (2012) 317.
[6] A. Biswas, G. Ebadi, M. Fessak, A. G. Johnpillai, S. Johnson, E. V. Krishnan and A. Yildirim, solutions of the perturbed Klein-Gordon equations, Iranian Journal of Science and Technology, Transaction A, 36( A4) (2012) 431-452.
[7] M. Dalir, M. Bashour, Applications of Fractional Calculus, , Appl. Math. Sci., 4(21) (2010), 1021-1032.
[8] J. F. Douglas, Some applications of fractional calculus to polymer science, Advances in chemical physics, 102 (2007) John Wiley and Sons, Inc.
[9] G. Ebadi, A. Yildirim and A. Biswas, chiral solitons with bohm potential using G'/G method and Exp-function method, , Romanian Reports in Physics 64(2) (2012) 357-366.
[10] Z. E. A. Fellah, C. Depollier, Application of fractional calculus to the sound waves propagation in rigid porous materials: Validation via ultrasonic measurement, Acta Acustica, 88 (2002) 34-39.
[11] R. W. Ibrahim, Fractional complex transforms for fractional di erential equations, Adv. Di erence Equ., 192 (2012) 1687-1847.
[12] H. Jafari, A. Sooraki, Y. Talebi, A. Biswas, The  rst integral method and traveling wave solutions to Davey{Stewartson equation, Nonlin. Anal.: Model. Control, 17 (2) (2012) 182-193.
[13] A. J. M. Jawad, M. D. Petkovic, and A. Biswas, Modi ed simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217(2) (2010) 869877.
[14] G. Jumarie, Fractional partial di erential equations and modi ed RiemannLiouville derivative, J. Appl. Math. Comput. 24(1-2) (2007) 31-48.
[15] K. Khan, M. Ali Akbar, Md. Nur Alam, Traveling wave solutions of the nonlinear DrinfeldSokolovWilson equation and modi ed BenjaminBonaMahony equations, journal of the Egyptian mathematical society, 21 (3) (2013) 233-240.
[16] K. Khan, M. Ali Akbar, Traveling wave solutions of nonlinear evolution equations via the enhanced (G/G)-expansion method, journal of the Egyptian mathematical society, 22(2) (2014) 220-226.
[17] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Di erential Equations, Elsevier, Amsterdam, (2006).
[18] E. V. Krishnan, G. Ebadi and A. Biswas, solutions of a coupled wave equation, University Politechnica of Bucharest Scienti c Bulletin, Series A., 75(2) (2013) 57-68.
[19] NA. Kudryashov, Exact solitary waves of the Fisher equation, Phys. Lett. A, 342(12) (2005) 99106.
[20] NA., Kudryashov, Simplest equation method to look for exact solutions of nonlinear di erential equations, Chaos Soliton Fract., 24(5) (2005) 1217-1231.
[21] V. Kulish and J. Lage,Application of Fractional Calculus to Fluid Mechanics, J. Fluids Eng., 124 (3), (2002) 803-806.
[22] Z.B. Li, J.H. He, Fractional Complex Transform for Fractional Di erential Equations, Math. Comput. Appl., 15(5) (2010) 970-973.
[23] Z.B. Li, J.H. He, Application of the Fractional Complex Transform to Fractional Di erential Equations, Nonlinear Sci. Lett.A, 2(3) (2011) 121-126.
[24] B. Lu, The  rst integral method for some time fractional di erential equations, J. Math. Anal. Appl. 395 (2) (2012) 684-693.
[25] R. L. Magin, Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus, Journal of Vibration and Control, 14(9-10) (2008) 1431-1442.
[26] S. Momania, Z. Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial di erential equations, Comp. Math. Appl., 54 (2007) 910-919.
[27] R. Morris, A.H. Kara, A. Chowdhury and A. Biswas, Soliton Solutions, Conservation Laws and Reductions of certain classes of Nonlinear Wave Equations, Zeitschrift fr Naturforschung A, 67a (2012) 613 -620.
[28] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23 (2002) 397-404.
[29] M. Timothy, Wave propagation in viscoelastic horns using a fractional calculus rheology model, Acoustical Society of America Journal, 114(4) (2003) 2442-2442.
[30] NK.Vitanov, ZI. Dimitrova, Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics, Commun. Nonlinear Sci. Numer. Simul. 15(10) (2010) 2836-2845.
[31] NK. Vitanov, ZI. Dimitrova, H. Kantz, Modi ed method of simplest equation and its application to nonlinear PDEs , Appl. Math. Compute. 216(9) (2010) 2587-2595.
[32] E. M. E. Zayed and S. A. H. Ibrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modi ed simple equation method, Chinese Physics Letters, 29( 6) (2012) article ID 060201.
Journal of Hyperstructures
Volume 9, Issue 1
June 2020
Pages 11-22

Files

Share

How to cite

Statistics