In this research, we study traveling wave solutions to the fractional extended nonlinear SchrÖdinger equation (NLSE), and the effects of the third-order dispersion parameter. This equation is used to simulate the propagation of femtosecond, plasma physic and in nonlinear optical fiber. To accomplish this goal, we use the extended simple equation approach and the improved F-expansion method to secure a variety of distinct solutions in the form of dark, singular, periodic, rational, and exponential waves. Also, the stability of the outcomes is effectively examined. Several graphs have been sketched under appropriate parametric values to reinforce some reported findings. Computational work along with a graphical demonstration confirms the exactness of the proposed methods. The issue has not previously been investigated by taking into account the impact of the third order dispersion parameter. The main objective of this study is to obtain the different kinds of traveling wave solutions of fractional extended NLSE which are absent in the literature which justify the novelty of this study. We believe that these novel solutions hold a prominent place in the fields of nonlinear sciences and optical engineering because these solutions will enables a through understanding of the development and dynamic nature of such models. The obtained results indicate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable traveling wave solutions to nonlinear equations emerging in various branches of scientific, technological, and engineering domains.

This study suggests a new approach for solving telegraph equation, commonly called damped wave equation, arising in electromagnetic waves and propagation of electrical signals. In this paper, He–Laplace method, formulated by He’s variational iteration method and Laplace transformation, is used to find the exact solution or a closed approximate solution of differential equations. The most distinct aspect of this method is that there is no need to calculate integration for next iterations in recurrence relations and convolution theorem is kept away to calculate the Lagrange multipliers in Laplace transformation. Moreover, He’s polynomials via homotopy perturbation method is introduced to bring down the computational work in nonlinear terms as Laplace transform has some limitation to nonlinear terms. The results obtained by proposed method indicate that this approach is easy to implement and converges rapidly to exact solution. Several problems are illustrated to demonstrate the accuracy and stability of this method.

This study is based on an analytical approach for nonlinear vibration of nonlinear evaluation equations which arises in shallow water waves. This approach is, coupled with variational iteration method and Laplace transformation, known as He–Laplace method. In order to show the capability of the proposed method, He’s polynomials are employed to handle nonlinear terms in illustrated problems due to specific limit of Laplace transformation. Significant results and their graphical representations demonstrated that He–Laplace method is most suitable for nonlinear problems, and it is enormously active for nonlinear vibrations and nonlinear waves.

Drinking kills a significant proportion of individuals every year, particularly in low‐income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low‐cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image of first aid in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method (GABMM).

The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable.

The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein–Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method ($\mathbb{Y}$HPTM), which is associated with the Yang transform ($\mathbb{Y}$T) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo–Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order ℘ values has been shown graphically. Our numerical investigations indicate that $\mathbb{Y}$HPTM is a simple and powerful mathematical tool to deal with the complexity of such problems.