Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method

Abstract

This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones.

Numerical solution of the Whitham-Broer-Kaup shallow water equation by quartic B-spline collocation method

“…into the linearized form of ( 14) and (15), where A and B are the harmonics amplitude, sh, f = s is the mode number, i 1 =and g is the amplification factor of the schemes, becomes…”

“…Due to the difficulty in obtaining exact solutions for the WBK shallow water equations, researchers have turned to approximate solutions using various analytical and numerical methods. Lately, numerous methods have been presented by many researchers for the solution of the coupled of nonlinear WBK problems such as [6] used the bifurcation method and qualitative theory of dynamical systems [7] applied homotopy analysis method [8] used homotopy Perturbation scheme [9] applied finite difference method by Radial Basis Function-Pseudospectral method [10] by variational iteration method (VIM) [11] applied Exp-function method [12] proposed differential transformation method [13] used Adomian decomposition method [14] utilized Yang transformation coupled with the Adomian technique [15] used hybrid approach to construct the approximate solution of the coupled nonlinear WBK and [16] proposed a time-space Chebyshev pseudo-spectral method. based on delta-shaped basis function is developed to solve generalized equal width equation [17,18].…”

Modified α -Parameterized Differential Transform Method for Solving Nonlinear Generalized Gardner Equation

“…The ease of use, accuracy of computations, and breadth of applications of DTM are its well-known benefits. Another significant benefit of this approach is its ability to drastically reduce the amount of computational labor required while still accurately delivering the series solution with a rapid convergence rate [3]. Many methods have been developed to solve nonlinear differential equations.…”