Abstract
In this paper, new algorithms called the “Modified exp(−Ω)-expansion function method” and “Improved Bernoulli sub-equation function method” have been proposed. The first algorithm is based on the exp(−Ω(ξ))-expansion method; the latter is based on the Bernoulli sub-Ordinary Differential Equation method. The methods proposed have been expressed comprehensively in this manuscript. The analytical solutions and application results are presented by drawing the two- and three-dimensional surfaces of solutions such as hyperbolic, complex, trigonometric and exponential solutions for the (2+1)-dimensional dispersive long water–wave system. Finally, a conclusion has been presented by mentioning the important discoveries in this manuscript.
Highlights
In recent years, studies on the nonlinear differential equations (NDEs) and systems (NDESs) have become a significant field among scientists
We have studied to obtain some new analytical solutions such as hyperbolic, exponential, and complex hyperbolic function solutions by applying modified exp( ́Ω(ξ))-expansion function method (MEFM) and Improved Bernoulli sub-equation function method (IBSEFM)
We have shown that these analytical solutions are verified in Equation (1)
Summary
Studies on the nonlinear differential equations (NDEs) and systems (NDESs) have become a significant field among scientists. This is why many engineering problems can be represented by using NDEs and with the rapid development of computational algorithms. One of them is to obtain various solutions of coastal, oceans and fluid problems such as approximate, numerical, analytical and traveling wave solutions. Equation (1) was used to model nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. Equation (1) appears in many scientific applications such as nonlinear fiber optics, plasma physics, fluid dynamics, and coastal engineering [16]
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