Abstract
The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.
Highlights
In the 20 century, nonlinear science (NLS) plays a significant role in special inventions, for example, the invention of the radio, the discovery of DNA structure for biology, the development of quantum theory for theoretical physics and chemistry, and the invention of transister for computer engineering
It is well known that NLS belongs to the nonlinear partial differential equations (NLPDEs) which are introduced in several areas such as fluid thermodynamics, plasma diffusion, biology, physics, geometry, and population dynamics
Lots of studies are focused on the differential equations [1,2,3,4,5,6,7,8,9,10], and many effective techniques are used to acquire analytical and numerical solutions for NLPDEs such as sine-cosine method [11], extended sinh-Gorden equation expansion method [12], variation iteration algorithm [13], homotopy perturbation method [14], F-expansion method [15], Exp-function expansion method [16], first integral method [17], Ansatz method [18], generalized Kudryashov method [19], ðG′/GÞ-expansion method [20], projective Riccati equation method [21], tanh method [22], nonpolynomial spline method [23], B-spline method [24], B-spline collocation [25], Weierstrass elliptic function method [26], Laplace decomposition method [27], extended direct algebraic method [28, 29], Sub-ODE method [30], Darboux transformation [31], and extended tanh-coth method [32, 33]
Summary
In the 20 century, nonlinear science (NLS) plays a significant role in special inventions, for example, the invention of the radio, the discovery of DNA structure for biology, the development of quantum theory for theoretical physics and chemistry, and the invention of transister for computer engineering. The generalized fifth-order KdV equation [34] is represented by wt + swwx + f w2wx + ewxxx + μwxxxxx = 0, ð1Þ where s, f , e, and μ are the arbitrary constants. This equation is a nonlinear model in many long wave physical phenomena. Dinarvand et al have found approximate analytical solutions of the sawada-kotera and Lax’s fifth-order KdV equations by homotopy analysis technique [36]. Salas and Lugo have introduced extended tanh method to obtain the exact solutions of the general fifth-order KdV equation [37].
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