Abstract
This paper employs an efficient technique, namely q-homotopy analysis transform method, to study a nonlinear coupled system of equations with Caputo fractional-time derivative. The nonlinear fractional coupled systems studied in this present investigation are the generalized Hirota–Satsuma coupled with KdV, the coupled KdV, and the modified coupled KdV equations which are used as a model in nonlinear physical phenomena arising in biology, chemistry, physics, and engineering. The series solution obtained using this method is proved to be reliable and accurate with minimal computations. Several numerical comparisons are made with well-known analytical methods and the exact solutions when alpha =1. It is evident from the results obtained that the proposed method outperformed other methods in handling the coupled systems considered in this paper. The effect of the fractional order on the problem considered is investigated, and the error estimate when compared with exact solution is presented.
Highlights
We consider the coupled systems: the generalized Hirota–Satsuma coupled with KdV, the coupled KdV, and the modified coupled KdV with Caputo fractional time derivative
The generalized Hirota– Satsuma coupled with KdV has been handled via different approaches such as the new iterative method (NIM) [35], homotopy perturbation method (HPM) [36], Adomian’ decomposition method (ADM) [37], homotopy analysis (HAM) [38], variational iteration method (VIM) [39], and reduced differential transformation method (RDTM) [40]
5 Numerical comparison This section is devoted to comparison of the results presented above with several other analytical methods in the literature such as new iterative method (NIM) in [35, 43], differential transformation method (DTM) and reduced differential transformation method
Summary
The study of fractional calculus, which involves fractional derivatives and integrals, has allured the interest of many in the field of engineering and natural sciences due to its monumental applications such as found in biotechnology [1], chaos theory [2], electrodynamics [3], random walk [4], signal and image processing [5, 6], nanotechnology [7], viscoelasticity [8], and other various fields [9,10,11,12,13,14,15,16,17,18]. Fan in [41] used an extended tanh-function method and symbolic computation to obtain four types of soliton solutions of the generalized Hirota–Satsuma coupled KDV and modified coupled KDV equations. Arife et al presented the numerical solutions of the generalized Hirota–Satsuma coupled KdV and modified coupled KDV equations through homotopy analysis method (HAM) [42]. Ganji et al in [44] used modified homotopy perturbation method to solve time-fractional generalized Hirota–Satsuma coupled KdV equations. 4 q-HATM application to a coupled system of time-fractional order We have carefully chosen a coupled system of strongly nonlinear time-fractional differential equations and have applied the q-HATM to obtain the analytical approximate solutions in the form of convergent series. + 1√ r7/2t3 cosh (x) – 5 tanh x sech x , 12 2 which as N → ∞ converges respectively to the exact solutions.
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