Abstract
In this paper, consider the eminent coupled Boussinesq–Burger (BB) equations and the coupled Whitham–Broer–Kaup (WBK) equations with time fractional derivative arising in the investigation of shallow water waves. The derivative is described in the sense of conformable derivative. We introduce the fundamental {(G'} / {G} )-expansion method and its extension, namely the two-variable {(G'} / {G}, {1} / {G} )-expansion method, to establish general solutions, some typical wave solutions existing in the literature, and some new and compatible soliton solutions comprised with certain parameters. For the definite values of these parameters, we derive and show in figures the well-known kink, singular kink, bell-shape soliton, periodic soliton, cuspon, and so on. The obtained solutions affirm that the introduced methods are reliable and efficient techniques to examine a wide variety of nonlinear fractional systems in the sense conformable derivative.
Highlights
The concept of fractional derivative is as old as that of the classical one, its advancement is not so old
Fractional differential and integral operators have eliminated the drawback of classical integer-order difficulties considering their nonlocal characteristics [4,5,6,7,8,9,10,11]
The Riemann–Liouville and Jumarie derivatives are recognized as a powerful modeling approach in the fields of viscoelasticity, viscoelastic deformation, viscous fluid [14,15,16], anomalous diffusion [17], and so on
Summary
The concept of fractional derivative is as old as that of the classical one, its advancement is not so old. Setting all coefficients of the polynomial to zero yields a set of algebraic equations, which can be solved with the help of Maple software package, and substituting the values of k, w, μ, λ, αi, βi into (4.10), we get analytical exact solutions to Eq (3.3) expressed by the hyperbolic function. If we accept B1 = 0, B2 = 0, we attain the subsequent singular periodic wave solutions to the nonlinear coupled Boussinesq–Burger equations: u21 (x, t). If we accept B2 = 0, B1 = 0, we determine the following singular periodic wave solutions to the nonlinear coupled BB equations: u22 (x, t). Equating each coefficient of this polynomial to zero, we obtain a system of algebraic equations, which is analyzed by applying Maple software package, and get the following results:
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