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This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering. [ABSTRACT FROM AUTHOR]

In this paper, the optical soliton and solitary wave solutions of the (2 + 1) -dimensional Mel'nikov equation are investigated using the Kudryashov R function technique. The Kudryashov R function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the (2 + 1) -dimensional Mel'nikov equation have been displayed in 3D, 2D and contour graphs. [ABSTRACT FROM AUTHOR]

In this paper, a reliable and effective local discontinuous Galerkin (LDG) scheme for numerically solving the classical Rosenau–Hyman equation with non-periodic boundary conditions has been proposed. This study employs the third-order nonlinearly stable total variation diminishing Runge–Kutta method and the LDG method, respectively, to discretize the temporal and spatial derivatives. Finally, numerical simulations are performed on various test problems and compared with the exact results as well as results produced by a few other numerical methods, to analyze the reliability and efficiency of the proposed method. The results generated, which validate the expected order of accuracy, are presented through multiple tables. In addition, several graphical representations of the problem are presented to depict the behavior of the solution. [ABSTRACT FROM AUTHOR]

The development of an effective and efficient local discontinuous Galerkin algorithm for the solution of the coupled (1 + 1) -dimensional Broer–Kaup system is the primary focus of this research. In this algorithm, spatial discretization is attained using the local discontinuous Galerkin method, while temporal discretization is handled using the explicit total variation diminishing higher-order Runge-Kutta method. However, the ( G ′ G ) -expansion method is also implemented to produce the exact solutions for the coupled (1 + 1) -dimensional Broer–Kaup system. The obtained solutions are the traveling wave solutions. The efficiency and reliability of the proposed method are analyzed by comparing the generated numerical simulations to various traveling wave solutions using several tables and figures. The exact solutions and the numerical simulations of the coupled (1 + 1) -dimensional Broer–Kaup system are shown to correspond exceptionally well. [ABSTRACT FROM AUTHOR]

Mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.