Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudrayshov method for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations, as a result of various steps, which on solving the so obtained equation systems yields the analytical solution. By this way various exact solutions including complex structures are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of solutions.


Introduction
In engineering and science, the problems arising from the wave propogation of communication between two (or) more systems such as electromagnetic waves in wireless sensor networks, water flow in dams during earthquake, stability of the output in current electricity, viscous flows in fluid dynamics, magneto hydro dynamics, turbulence in microtides, and other physical phenomenons are described by the non-linear evolution equations (NLEE). The process of solving such NLEE analytically and numerically uses symbolic computation procedures, analytical methods and cardinal functions respectively. In modelling such media continuously takes to the Generalized Kuramoto-Sivashinsky Equation (GKSE) [1] given by the partial differential equation nonlinearly for u = u(x,t) and non-zero α , β , γ constants.
GKSE and its solutions performs ample roles on flowing in viscous fluids, feedback in the output of self loop controllers, trajectories systems, gas dynamics. While α = γ = 1 and β = 0, GKSE Eq (1) leads to Kuramoto-Sivashinsky Equation (KSE). N. A. Kudryashov solved Eq (1) by the method of Weiss-Tabor-Carnevale and obtained exact solutions in [1]. E. J. Parkes et al applied tanh method for Eq (1) by taking α = β = 1 and solved using Mathematica package, they also solved (1) by taking α = −1 , β = 1 in [2]. B. Abdel-Hamid in [3] assumed the inital solution as PDE for u and solved exactly for α = 1 , β = 0 in Eq (1). D. Baldwin et al [4] applied tanh and sech methods to Eq (1) with α = γ = 1 and solved using mathematica package. C. Li et al [5] solved GKSE of the form u t + β u α u x + γu τ u xx + δ u xxxx = 0 using Bernoulli equation. By simplest equation method again N. A. Kudrayshov solved by considering u x = u m u x in GKSE Eq (1) and obtained solution for general m with some restrictions in [6].
A. H. Khater et al in [7] used Chebyshev polynomials and applied its collocation points to solve approximations of Eq (1). M. G. Porshokouhi et al in [8] [11] where they used tanh exact solutions for error estimations. J. Yang et al in [12] used sine-cosine method and dynamic bifurcation method to solve more generalized GKSE and its related equations to Eq (1). In [12] J. Rashidinia et al solved Eq (1) by Chebyshev wavelets. O.Acan et al applied reduced differential transform method to solve Eq (1) by taking β = 0 in [14]. For solving the nonlinear partial differential equations there as been many schemes applied such as Kudryashov method by M. Foroutan et al in [15] and K. K. Ali et al in [16]. Modified Kudryashov method by K. Hosseini et al in [17,18], D. Kumar et al in [19], A. K. Joardar et al in [20] and A.R. Seadawy et al in [21]. Generalized Kudryashov method by F. Mahmud et al in [22], S. T. Demiray et al in [23] and S. Bibi et al in [24]. Sine-cosine method by K. R. Raslan et al in [25]. Sine-Gordon method by H. Bulut et al in [26]. Sineh-Gordon equation expansion method by H. M. Baskonus et al in [27], Y. Xian-Lin et al in [28] and A. Esen et al in [29]. Extended trial equation method by K. A. Gepreel in [30],Y. Pandir et al in [31] and Y. Gurefe et al in [32]. Exponential − φ 2 method by L.K. Ravi et al in [33], A. R. Seadawy et al in [34] and M. Nur Alam et al in [35]. Jacobi elliptic function method by S. Liu et al in [36]. F-expansion method by A. Ebaid et al in [37].

Analysis of the Modified Kudrayshov Method
Given the nonlinear partial differential equations (NLPDE) which are converted to the ordinary differential equations (ODE) by making the necessary transformation. Then the initially assumed solution is substituted in the ODE, from which the algebraic equations are obtained and solved for unknowns, substituting the obtained unknown values in the assumed solution gives the exact solution of NLPDE. MKM takes the following steps in solving NLPDE [17][18][19][20][21].
Step 1. Consider the given NLPDE of the following form u = u(x,t) . (2) Step 2. Apply the wave transformation u(x,t) = u(η) in Eq (2) where Here µ is the wave variable and λ is the velocity, both are non-zero constants. Hence Eq (2) transforms to the following ODE.
where the prime represents the derivative w. r. t. η.
Step 3. Let the initial solution guess of Eq (4) be, where N is non-zero and positive constant calculated by principle of homogeneous balancing of Eq (4), A i ; i = 0, 1, 2, · · · are unknowns to be calculated and Q(η) is the solution of the following auxilary ODE.
given by, where D is the integral constant and assumed D = 1.
Step 5. Finally substituting the values of Step 4 in Eq (5) and then in Eq (3) gives the solution u(x,t) of Eq (2).

Applications to solve Generalized Kuramoto-Sivashinsky equation
Applying the wave transformation with Eq (3) to the GKSE Eq (1) leads to the ODE and then integrating once the ODE by taking integration constant to zero, transforms to the following ODE.
Case 18. For α = −δ 2 and β = 4iδ 1 in Eq (1), the unknown coeffecients are given by, Therefore the exact complex solution of Eq (1) is given by, The 3D complex graph of real and imaginary parts of u 18 (x,t) for a = 7 and µ = γ = 1.5 are drawn in Figure 2.
Case 19. For α = −δ 2 and β = −4iδ 1 in Eq (1), the unknown coeffecients are given by, Therefore the exact complex solution of Eq (1) is given by, The 3D complex graph of real and imaginary parts of u 19 (x,t) for a = 7 and µ = γ = 1.5 are drawn in Figure 3.
Case 20. For α = −δ 2 and β = −4iδ 1 in Eq (1), the unknown coeffecients are given by, Therefore the exact complex solution of Eq (1) is given by, The 3D complex graph of real and imaginary parts of u 20 (x,t) for a = 7 and µ = γ = 1.5 are drawn in Figure 4.

Conclusion
In this work the generalized Kuramoto-Sivashinsky equation is solved for exact solutions. The said GKSE have exact solutions for the different α and β values, which we obtained by the application of modified Kudrayshov method and found 10 classes of (α , β ) pairs and their corresponding two distinct exact solutions for each class of GKSE Eq (1) from cases 1 through 20. All the solutions are validated in Maple computer algebra. The three dimensional simulations of solutions shows their behavioural pattern. We reckon all the solutions obtained through this communication will help further study of GKSE in the physical field.