## The time-fractional coupled-Korteweg-de-Vries equations.(English)Zbl 1291.35273

Summary: We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35R11 Fractional partial differential equations
Full Text:

### References:

 [1] Kurulay, M.; Secer, A.; Akinlar, M. A., A new approximate analytical solution of Kuramoto-Sivashinsky equation using homotopy analysis method, Applied Mathematics & Information Sciences, 7, 1, 267-271, (2013) [2] Oldham, K. B.; Spanier, J., The Fractional Calculus. The Fractional Calculus, Mathematics in Science and Engineering, 111, xiii+234, (1974), New York, NY, USA: Academic Press, New York, NY, USA [3] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008 [4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, xvi+523, (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003 [5] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008 [6] Caputo, M., Linear models of dissipation whose Q is almost frequency independent—II, Geophysical Journal International, 13, 5, 529-539, (1967) [7] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, xvi+523, (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003 [8] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, xvi+366, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002 [9] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, xxxvi+976, (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003 [10] Zaslavsky, G. M., Hamiltonian Chaos and Fractional Dynamics, xiv+421, (2008), Oxford, UK: Oxford University Press, Oxford, UK · Zbl 1152.37001 [11] Atangana, A., Numerical solution of space-time fractional derivative of groundwater flow equation, International Conference of Algebra and Applied Analysis [12] Hirota, R.; Satsuma, J., Soliton solutions of a coupled Korteweg-de Vries equation, Physics Letters A, 85, 8-9, 407-408, (1981) [13] Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 27-34, (2006) · Zbl 1401.65087 [14] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, xvi+366, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002 [15] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008 [16] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, xxxvi+976, (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003 [17] Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to $$(\text{d} t)^a,$$ Applied Mathematics Letters, 18, 7, 739-748, (2005) · Zbl 1082.60029 [18] Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers & Mathematics with Applications, 51, 9-10, 1367-1376, (2006) · Zbl 1137.65001 [19] Atangana, A.; Botha, J. F., Analytical solution of the groundwater flow equation obtained via homotopy decomposition method, Journal of Earth Science & Climatic Change, 3, 2, 115, (2012) [20] Merdan, M.; Mohyud-Din, S. T., A new method for time-fractionel coupled-KDV equations with modified Riemann-Liouville derivative, Studies in Nonlinear Science, 2, 2, 77-86, (2011) [21] El-Wakil, S. A.; Abulwafa, E. M.; Zahran, M. A.; Mahmoud, A. A., Time-fractional KdV equation: formulation and solution using variational methods, Nonlinear Dynamics, 65, 1-2, 55-63, (2011) · Zbl 1234.35219 [22] He, J.-H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 3-4, 257-262, (1999) · Zbl 0956.70017 [23] Matinfar, M.; Ghanbari, M., The application of the modified variational iteration method on the generalized Fisher’s equation, Journal of Applied Mathematics and Computing, 31, 1-2, 165-175, (2008) · Zbl 1216.65138 [24] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Communications in Nonlinear Science and Numerical Simulation, 13, 3, 539-546, (2008) · Zbl 1132.34305 [25] Biazar, J., Solving system of integral equations by Adomian decomposition method [Ph.D. thesis], (2002), Iran: Teacher Training University, Iran · Zbl 1012.65146 [26] Atangana, A., New class of boundary value problems, Information Sciences Letters, 1, 2, 67-76, (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.