Abstract:
In this work, the auxiliary equation method is applied to derive exact solutions of nonlinear fractional Klein–Gordon equation and space-time fractional Symmetric Regularized Long Wave equation. Consequently, some exact solutions of these equations are successfully obtained. These solutions are formed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions founded by the suggested method indicate that the approach is easy to implement and powerful.
Acknowledgements:
The authors Arzu Akbulut and Melike Kaplan are grateful to The Scientific and Technological Research Council of Turkey for granting scholarship for PhD studies.
References
[1] K. B. Oldham and F. Spanier, The fractional calculus, Academic Press, New York, 1974.Suche in Google Scholar
[2] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.Suche in Google Scholar
[3] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.Suche in Google Scholar
[4] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.Suche in Google Scholar
[5] C. Li and F. Zeng, Numerical methods for fractional calculus, Chapman and Hall/CRC, Boca Raton, 2015.10.1201/b18503Suche in Google Scholar
[6] Z. Odibat and S. Momanib, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl. 58 (2009), 2199–2208.10.1016/j.camwa.2009.03.009Suche in Google Scholar
[7] A. Bekir, O. Guner and A.C. Cevikel, The fractional complex transform and exp-function methods for fractional differential equations, Abstr. Appl. Anal. 2013 (2013), 426462.10.1155/2013/426462Suche in Google Scholar
[8] H. Jafari, S. Das and H. Tajadodi, Solving a multi-order fractional differential equation using homotopy analysis method, J. King Saud Univ. Sci. 23 (2011), 151–155.10.1016/j.jksus.2010.06.023Suche in Google Scholar
[9] N. Shang and B. Zheng, Exact solutions for three fractional partial differential equations by the (G′/G) method, Int. J. Appl. Math. 43 (3) (2013), 114–119.10.1186/1687-1847-2013-199Suche in Google Scholar
[10] K. Khan, M. A. Akbar, M. M. Rashidi and I. Zamanpour, Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method, Waves Random Complex Media 25 (4) (2015), 644–655.10.1080/17455030.2015.1068964Suche in Google Scholar
[11] Islam, Md.S., K. Khan and M.A. Akbar, An analytical method for finding exact solutions of modified Korteweg-de Vries equation, Results Phys. 5 (2015), 131–135.10.1016/j.rinp.2015.01.007Suche in Google Scholar
[12] A. Bekir, O. Guner and B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl. 395 (2012), 684–693.10.1016/j.jmaa.2012.05.066Suche in Google Scholar
[13] M. Eslami, B. F. Vajargah, M. Mirzazadeh and A. Biswas, Application of first integral method to fractional partial differential equations. Indian J. Phys. 88 (2) (2014), 177–184.10.1007/s12648-013-0401-6Suche in Google Scholar
[14] B. Tong, Y. He, L. Wei and X. Zhang, A generalized fractional sub-equation method for fractional differential equations with variable coefficients, Phys. Lett. A 376 (3) (2012), 2588–2590.10.1016/j.physleta.2012.07.018Suche in Google Scholar
[15] J. F. Alzaidy, Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics, Brit. J. Math. Comput. Sci. 3 (2) (2013), 153–163.10.9734/BJMCS/2013/2908Suche in Google Scholar
[16] W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana J. Phys. 81 (3) (2013), 377–384.10.1007/s12043-013-0583-7Suche in Google Scholar
[17] M. Kaplan, A. Bekir, A. Akbulut and E. Aksoy, The modified simple equation method for nonlinear fractional differential equations, Romanian J. Phys. 60 (9–10)(2015), 1374–1383.Suche in Google Scholar
[18] H. Bulut and Y. Pandir, Modified trial equation method to the nonlinear fractional Sharma-Tasso-Olever equation, Int. J. Model. Optim. 3 (4) (2013), 353–357.10.7763/IJMO.2013.V3.297Suche in Google Scholar
[19] Md. S. Islam, K. Khan and M. A. Akbar, Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations, SpringerPlus. 4 124 (2015).10.1186/s40064-015-0893-ySuche in Google Scholar PubMed PubMed Central
[20] S. Sirendaoreji, A new auxiliary equation and exact travelling wave solutions of nonlinear equations, Phys. Lett. A 356 (2006), 124–130.10.1016/j.physleta.2006.03.034Suche in Google Scholar
[21] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica 24 (2) (1999), 207–233.Suche in Google Scholar
[22] J. Liao, F. Chen and S. Hu, Existence of solutions for fractional impulsive neutral functional differential equations with in finite delay, Neurocomputing 122 (2013), 156–162.10.1016/j.neucom.2013.06.034Suche in Google Scholar
[23] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (2006), 1367–1376.10.1016/j.camwa.2006.02.001Suche in Google Scholar
[24] Z. B. Li and J. H.He, Fractional complex transform for fractional differential equations, Math. Comput. Appl. 15 (2010), 970–973.10.3390/mca15050970Suche in Google Scholar
[25] J. H. He and Z. B. Li, Converting fractional differential equations into partial differential equations. Therm Sci. Math. Comput. 16 (2) (2012), 331–334.10.2298/TSCI110503068HSuche in Google Scholar
[26] S. Zhang and T. Xia, A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Phys. Lett. A 363 (2007), 356–360.10.1016/j.physleta.2006.11.035Suche in Google Scholar
[27] H. Jafari, H. Tajadodi, N. Kadkhoda and D. Baleanu, Fractional sub-equation method for Cahn-Hilliard and Klein-Gordon equations, Abstr. Appl. Anal. 2013 (2013), 587179.Suche in Google Scholar
[28] A. K. Golmankhaneh, A. Golmankhaneh and D. Baleanu, On nonlinear fractional Klein–Gordon equation, Signal Process. 91 (2011), 446–451.10.1016/j.sigpro.2010.04.016Suche in Google Scholar
[29] R. Garra, E. Orsingher and F. Polito, Fractional Klein–Gordon equations and related stochastic processes, J. Stat. Phys. 155 (2014), 777–809.10.1007/s10955-014-0976-0Suche in Google Scholar
[30] A. Biswas, C. Zony and E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation, Appl. Math. Comput. 203 (2008), 153–156.10.1016/j.amc.2008.04.013Suche in Google Scholar
[31] R. Sassaman and A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations, Commun Nonlinear Sci Numer Simul. 14 (2009), 3239–3249.10.1016/j.cnsns.2008.12.020Suche in Google Scholar
[32] C. M. Khalique and A. Biswas, Analysis of nonlinear Klein–Gordon equations by lie symmetry, Appl. Math. Lett. 23 (2010), 1397–1400.10.1016/j.aml.2010.07.006Suche in Google Scholar
[33] R. Sassaman, A. Heidari and A. Biswas, Topological and non-topological solitons of nonlinear Klein–Gordon equations by He’s semi-inverse variation principle, J. Franklin Inst. 347 (2010), 1148–1157.10.1016/j.jfranklin.2010.04.012Suche in Google Scholar
[34] A. Biswas, M. Song and E. Zerrad, Bifurcation analysis and implicit solutions of Klein–Gordon equation with dual-power law nonlinearity in relativistic quantum mechanics, Int J Nonlinear Sci. Numer. Simul. 14 (2013), 317–322.10.1515/ijnsns-2013-0040Suche in Google Scholar
[35] J. F. Alzaidy, The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs, Am. J. Math. Anal. 1 (1) (2013), 14–19.Suche in Google Scholar
[36] N. Taghizadeh, M. Mirzazadeh, M. Rahimian and M. Akbari, Application of the simplest equation method to some time-fractional partial differential equations, Ain Shams Eng. J. 4 (2013), 897–902.10.1016/j.asej.2013.01.006Suche in Google Scholar
[37] W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Indian Acad. Sci. 81 (2013), 377–384.10.1007/s12043-013-0583-7Suche in Google Scholar
©2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Existence of Solutions of a New Class of Impulsive Initial Value Problems of Singular Nonlinear Fractional Differential Systems
- Lump-Type Solutions to the (3+1)-Dimensional Jimbo-Miwa Equation
- Comparison of Subcritical and Supercritical Flow Patterns Within Triangular Channels Along the Side Weir
- Existence of Solutions for Schrödinger–Kirchhoff Type Problems Involving Nonlocal Elliptic Operators
- Application of Fractional Techniques in the Analysis of Forest Fires
- Discharge Coefficient of Rectangular Side Weirs on Circular Channels
- Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate
- Auxiliary Equation Method for Fractional Differential Equations with Modified Riemann–Liouville Derivative
Artikel in diesem Heft
- Frontmatter
- Existence of Solutions of a New Class of Impulsive Initial Value Problems of Singular Nonlinear Fractional Differential Systems
- Lump-Type Solutions to the (3+1)-Dimensional Jimbo-Miwa Equation
- Comparison of Subcritical and Supercritical Flow Patterns Within Triangular Channels Along the Side Weir
- Existence of Solutions for Schrödinger–Kirchhoff Type Problems Involving Nonlocal Elliptic Operators
- Application of Fractional Techniques in the Analysis of Forest Fires
- Discharge Coefficient of Rectangular Side Weirs on Circular Channels
- Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate
- Auxiliary Equation Method for Fractional Differential Equations with Modified Riemann–Liouville Derivative