Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity

Abstract

This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

A discretization-free deep neural network-based approach for advection-dispersion-reaction mechanisms

“…Even if they present acceptable, accurate results, the challenging points, low computational cost, and catching sensitive behaviours protect their importance. Recently, the thirdorder dispersive PDE possessing two travelling wave solutions has been explored in literature [29], considering also the nonlinearity and various parameters in equation (1) such as n, ò, μ, and r. It has a wide range of applicability due to the flexibility of defining the given parameters.…”

“…and proceed by approximating u(x, t) by a DNN, taking into consideration physics-informed neural networks. Figures 4, 5, 6 and 7 demonstrate the results of the current applications for different sets of parameters compared with travelling wave solutions in [29]. Figures 8 and 9 show the compared results.…”

“…Let us consider the third-order dispersive PDE, equation (2), including nonlinearity with the parameters in table 1 which has been taken into consideration with the space and time conditions x ä [ − 40, 40], t ä [0, 10] and its two travelling wave solutions mentioned in the previous section [29]. For comparison purposes, the implementations have been produced in MATLAB 2023b, which is installed on a computer that has the properties of a 2.80 GHz 11th Gen Intel Core i7 and 16 GB of RAM.…”

M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation

“…Because of its wide application, investigation of the analytical and soliton solutions of the NLPDEs with integer or fractional order has been very popular among authors over the past few decades. Numerous techniques consisting of the analytical and numerical methods have been improved to gain the soliton and analytical solutions of the PDEs such as the combined improved Kudryashov-new extended auxiliary sub equation method [1], the enhanced modified extended tanh-expansion approach [2,3], the sine-Gordon equation approach [4], F-expansion method [5], the tanh-coth function, the modified kudryashov expansion and rational sine-cosine approaches [6], the Riccati equation method [7], the tan(Θ/2) expansion approach [8], the Jacobi elliptic functions methodology [9], the generalized Bernoulli sub-ODE scheme [10], the extended ( G ′ G 2 )-expansion scheme [11], Nucci's reduction method [12], the new Kudryashov method [13][14][15], the sub-equation method based on Riccati equation [16], and the modified Sardar subequation method [17].…”