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Articles

Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative

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Pages 678-693 | Received 19 Jun 2017, Accepted 05 Apr 2018, Published online: 04 May 2018

References

  • Baleanu D, Diethelm K, Scalas E, et al. Fractional calculus models and numerical methods. Series on complexity, nonlinearity and chaos. Singapore, World Scientific; 2012.
  • Petras I. Fractional-order nonlinear systems: modeling, analysis and simulation. New York, Springer Science & Business Media; 2011.
  • Zhang Y, Li Z, Gao J, et al. A method for designing assembly tolerance networks of mechanical assemblies. Math Prob Eng. 2012;2012:1–27.
  • Podlubny I. Fractional differential equations: an introduction to fractional derivatives, Fractional differential equations, to methods of their solution and some of their applications. San Diego, Calif, USA: Academic Press; 1999.
  • Zhai Y, Liu L, Lu W, et al. The application of disturbance observer to propulsion control of sub-mini underwater robot. International Conference on Computational Science and its Applications. Berlin: Springer; 2010. p. 590–598.
  • Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam, The Netherlands: Elsevier Science B.V; 2006.
  • Choo KY, Muniandy SV, Woon KL, et al. Modeling anomalous charge carrier transport in disordered organic semiconductors using the fractional drift-diffusion equation. Org Electron. 2017;41:157–165.
  • Abbas M, Rizvi AA, Naqvi QA. Fractional dual fields to the Maxwell equations for a line source buried in dielectric half space. Optik-Int J Light Electron Opt. 2017;129:225–230.
  • Formato A, Ianniello D, Villecco F, et al. Design optimization of the plough working surface by computerized mathematical model. Emirates J Food Agric. 2017;1:36–44.
  • Tariq H, Akram G. New approach for exact solutions of time fractional Cahn-Allen equation and time fractional Phi-4 equation. Physica A. 2017;473:352–362.
  • Pellegrino A, Villecco F. Design optimization of a natural gas substation with intensification of the energy cycle. Math Prob Eng. 2010;2010:1–11.
  • Wang L, Sun DA, Li P, et al. Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils. Comput Geotechnics. 2017;83:30–39.
  • Oskouie MF, Ansari R. Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects. Appl Math Model. 2017;43:337–350.
  • Odibat ZM, Kumar S, Shawagfeh N, et al. A study on the convergence conditions of generalized differential transform method. Math Methods Appl Sci. 2017;40(1):40–48.
  • Unal E, Gokdogan A. Solution of conformable fractional ordinary differential equations via differential transform method. Optik-Int J Light Electron Opt. 2017;128:264–273.
  • Shah K, Singh T, Kilicman A. Combination of integral and projected differential transform methods for time-fractional gas dynamics equations. Ain Shams Eng J. 2017;1:1–6.
  • Mohyud-Din ST, Noor MA, Noor KI, et al. Variational iteration method for re-formulated partial differential equations. Int J Nonlinear Sci Numer Simul. 2010;11(2):87–92.
  • Goswami P, Alqahtani RT. On the solution of local fractional differential equations using local fractional Laplace variational iteration method. Math Prob Eng. 2016;2016:1–6.
  • Jafari H, Jassim HK. Numerical solutions of telegraph and laplace equations on cantor sets using local fractional Laplace decomposition method. Int J Adv Appl Math Mech. 2015;2(3):144–151.
  • Tauseef Mohyud-Din S, Yildirim A, AnilSezer S. Numerical soliton solutions of improved Boussinesq equation. Int J Numer Methods Heat Fluid Flow. 2011;21(7):822–827.
  • Mohyud-Din ST, Noor MA, Noor KI. Some relatively new techniques for nonlinear problems. Math Prob Eng. 2009;2009:1–25.
  • Tauseef Mohyud-Din S, Yildirim A, Sariaydin S. Numerical soliton solution of the Kaup-Kupershmidt equation. Int J Numer Methods Heat Fluid Flow. 2011;21(3):272–281.
  • Ma HC, Yao DD, Peng XF. Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method. Thermal Sci. 2015;19(4):1239–1244.
  • Feng D, Li K. Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method. Phys Lett A. 2011;375(23):2201–2210.
  • Mohyud-Din ST, Bibi S. Exact solutions for nonlinear fractional differential equations using exponential rational function method. Opt Quantum Electron. 2017;49(2):1–12.
  • Mohyud-Din ST, Irshad A. Solitary wave solutions of some nonlinear PDEs arising in electronics. Opt Quantum Electron. 2017;49(4):1–12.
  • Tauseef Mohyud-Din S, Khan Y, Faraz N, et al. Exp-function method for solitary and periodic solutions of Fitzhugh-Nagumo equation. Int J Numer Methods Heat Fluid Flow. 2012;22(3):335–341.
  • Noor MA, Mohyud-Din ST, Waheed A. Exp-function method for generalized traveling solutions of master partial differential equation. Acta Appl Math. 2008;104(2):131–137.
  • Mohyud-Din ST, Noor MA, Waheed A. Exp-function method for generalized travelling solutions of Calogero-Degasperis-Fokas equation. Z Naturforsch A. 2010;65(1–2):78–84.
  • Noor MA, Mohyud-Din ST, Waheed A, et al. Exp-function method for traveling wave solutions of nonlinear evolution equations. Appl Math Comput. 2010;216(2):477–483.
  • Tauseef Mohyud-Din S, Negahdary E, Usman M. A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations. Int J Numer Methods Heat Fluid Flow. 2012;22(5):641–658.
  • Alam MN, Akbar MA, Mohyud-Din ST. A novel (G’/G)-expansion method and its application to the Boussinesq equation. Chinese Phys B. 2013;23(2):1–10.
  • Mohyud-Din ST, Noor MA, Noor KI. Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials. Int J Nonlinear Sci Numer Simul. 2009;10(2):227–234.
  • Ma WX, You Y. Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals. 2004;22(2):395–406.
  • Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl. 2010;59(3):1326–1336.
  • Zhou Y, Jiao F, Li J. Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal. 2009;71(7–8):2724–2733.
  • Zhang S, Zhang HQ. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys Lett A. 2011;375(7):1069–1073.
  • Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput Math Appl. 2006;51:1367–1376.
  • Alzaidy JF. The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. Am J Math Anal. 2013;1(1):14–19.
  • Wu Y, Geng X, Hu X, et al. A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformations. Phys Lett A. 1999;255(4–6):259–264.
  • Fan E. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation. Phys Lett A. 2001;282(1–2):18–22.
  • Khalil R, Al Horani M, Yousef A, et al. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70.
  • Cenesiz Y, Kurt A. The solution of time fractional heat equation with new fractional derivative definition. 8th International Conference on Applied Mathematics, Simulation, Modelling (Italy, ASM 2014). 2014; p. 195–198.
  • Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math. 2015;13(1):1–10.
  • Cenesiz Y, Baleanu D, Kurt A, Tasbozan O. New exact solutions of Burgers’ type equations with conformable derivative. Waves Random Complex Media. 2016;1:1–14.
  • He S, Sun K, Mei X, et al. Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative. Eur Phys J Plus. 2017;132(1):1–11.
  • Cenesiz Y, Kurt A. The new solution of time fractional wave equation with conformable fractional derivative definition. J New Theor. 2015;7:79–85.
  • Rezazadeh H, Ziabarya BP. Sub-equation method for the conformable fractional generalized kuramoto sivashinsky equation. Comput Res Progress App Sci Eng. 2016;2(3):106–109.
  • Aminikhah H, Sheikhani AR, Rezazadeh H. Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives. Sci Iranica Trans B Mech Eng. 2016;23(3):1048.
  • Hosseini K, Mayeli P, Ansari R. Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Optik-Int J Light Electron Opt. 2017;130:737–742.
  • Cenesiz Y, Tasbozan O, Kurt A. Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system. Tbilisi Math J. 2017;10(1):117–125.
  • Atangana A, Baleanu D, Alsaedi A. Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Phys. 2016;14(1):145–149.
  • Garshasbi M, Momeni F. Numerical solution of Hirota-Satsuma coupled mKdV equation with quantic B-spline collocation method. J Comput Sci Comput Math. 2011;2011:13–18.
  • Liu J, Li H. Approximate analytic solutions of time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. Abstr Appl Anal. 2013;2013:1–11.
  • Hirota R, Satsuma J. Soliton solutions of a coupled Korteweg-de Vries equation. Phys Lett A. 1981;85(8–9):407–408.

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