On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients

Abstract

In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.

Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity

“…where p i , q i , c i , α , and αs are arbitrary constants to be determined later. It is worth to be noticing that, n, r and k are determined from the balance equation by the criteria given in [39][40][41][42][43][44][45]. Also, a second condition (the consistency condition), which asserts that the arbitrary functions in Eq.…”

“…by using the uni ed method. The uni ed method (UM) [39][40][41][42][43][44][45] and its generalized form (GUM) [46][47][48][49][50][51], the generalized uni ed method, that introduce a simple algorithm to nd the exact solutions and multi-wave solutions respectively in nonlinear evolution equations and nonlinear conformable fractional evolution equations both with constant and variable coecients.…”

Analytical and numerical solutions of mathematical biology models: The Newell‐Whitehead‐Segel and Allen‐Cahn equations

“…Here, the key ideas of the unified method [27][28][29][30][31][32][33][34][35][36] for extracting some analytical wave solutions for Equation 1 are described.…”

“…To show analytical wave solution of Equation 32, the parameters are chosen as a = 6 2 , b = 0.5, = 0.2, = 0.8 and c = 5 . The computations are done up to t = 100 for these parameters.…”

“…Clearly, it is seen from the Table 9 that error norms increase as values of Δt decrease at bigger times. The Figure 6A is presented numerical solutions of Equation 32 for Δx = 0.1 and Δt = 10 −4 . Also, absolute errors are shown in Figure 6B for Δx = 0.01 and Δt = 10 −5 at selected times.…”