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The extended fractional subequation method for nonlinear fractional differential equations. (English) Zbl 1264.35272
Summary: An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansatz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
35R11Fractional partial differential equations
35C05Solutions of PDE in closed form
35Q53KdV-like (Korteweg-de Vries) equations
Full Text: DOI
[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000. · Zbl 1177.26011 · doi:10.1142/9789812817747
[3] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. · Zbl 1225.62144 · doi:10.1111/j.0006-341X.2003.00121.x
[4] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[5] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0924.44003 · doi:10.1080/10652469308819017
[6] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[7] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[8] V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994. · Zbl 0894.26002
[9] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[10] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007. · Zbl 1125.93029 · doi:10.1007/978-1-4020-6042-7_37
[11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. · Zbl 1222.26015 · doi:10.1142/9781848163300 · eudml:219561
[12] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA, 2012. · Zbl 1248.26011 · doi:10.1142/9789814355216
[13] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong.
[14] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
[15] A. H. A. Ali, “The modified extended tanh-function method for solving coupled MKdV and coupled Hirota-Satsuma coupled KdV equations,” Physics Letters. A, vol. 363, no. 5-6, pp. 420-425, 2007. · Zbl 1197.35212 · doi:10.1016/j.physleta.2006.11.076
[16] C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352-3368, 2011. · Zbl 1218.65070 · doi:10.1016/j.jcp.2011.01.030
[17] G. H. Gao, Z. Z. Sun, and Y. N. Zhang, “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions,” Journal of Computational Physics, vol. 231, no. 7, pp. 2865-2879, 2012. · Zbl 1242.65160 · doi:10.1016/j.jcp.2011.12.028
[18] W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204-226, 2008/09. · doi:10.1137/080714130
[19] S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379-387, 2007. · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[20] Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194-199, 2008. · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[21] Y. Hu, Y. Luo, and Z. Lu, “Analytical solution of the linear fractional differential equation by Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 220-229, 2008. · Zbl 1132.26313 · doi:10.1016/j.cam.2007.04.005
[22] A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters. A, vol. 359, no. 3, pp. 175-182, 2006. · Zbl 1236.35003 · doi:10.1016/j.physleta.2006.06.024
[23] A. M. A. El-Sayed, S. H. Behiry, and W. E. Raslan, “Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1759-1765, 2010. · Zbl 1189.35358 · doi:10.1016/j.camwa.2009.08.065
[24] Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2199-2208, 2009. · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[25] M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476-484, 2008. · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[26] G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters. A, vol. 374, no. 25, pp. 2506-2509, 2010. · Zbl 1237.34007 · doi:10.1016/j.physleta.2010.04.034
[27] J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[28] E. Fan, “Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation,” Physics Letters. A, vol. 282, no. 1-2, pp. 18-22, 2001. · Zbl 0984.37092 · doi:10.1016/S0375-9601(01)00161-X
[29] J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[30] S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters. A, vol. 375, no. 7, pp. 1069-1073, 2011. · Zbl 1242.35217 · doi:10.1016/j.physleta.2011.01.029
[31] M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters. A, vol. 199, no. 3-4, pp. 169-172, 1995. · Zbl 1020.35528 · doi:10.1016/0375-9601(95)00092-H
[32] G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367-1376, 2006. · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[33] G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31-48, 2007. · Zbl 1145.26302 · doi:10.1007/BF02832299
[34] S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters. A, vol. 376, no. 4, pp. 407-411, 2012. · Zbl 1255.37022 · doi:10.1016/j.physleta.2011.10.056
[35] B. Lu, “Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters. A, vol. 376, no. 28-29, pp. 2045-2048, 2012. · Zbl 1266.35139 · doi:10.1016/j.physleta.2012.05.013
[36] G. Jumarie, “Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1444-1450, 2010. · Zbl 1202.30068 · doi:10.1016/j.aml.2010.08.001
[37] J. H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters. A, vol. 376, no. 4, pp. 257-259, 2012. · Zbl 1255.26002 · doi:10.1016/j.physleta.2011.11.030
[38] J. H. He, “Asymptotic methods for solitary solutions and compacts,” Abstract and applied analysis. In press. · doi:10.1155/2012/916793
[39] S. E. Esipov, “Coupled Burgers equations: a model of polydispersive sedimentation,” Physical Review E, vol. 52, no. 4, pp. 3711-3718, 1995. · doi:10.1103/PhysRevE.52.3711
[40] A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Physica A, vol. 361, no. 2, pp. 394-404, 2006. · doi:10.1016/j.physa.2005.07.008