The Zakharov equation is a nonlinear plasma fluid model, used for ion-acoustic waves in a magnetized plasma. In the present study, Langmuir waves of the dimensionless Zakharov equation are investigated by using the Sardar-subequation method. The obtained solutions lead to a variety of exact solutions in the form of dark, bright, periodic singular,  singular, and combined dark-bright type solutions. These acquired solutions are depicted graphically by the 2D, contour and 3D plots which show the physical behaviour of obtained solutions. All the graphs confirm the validity of the obtained solutions. These types of solutions have a large range of applications in mathematical and applied sciences.

DNA, or deoxyribonucleic acid, is found in every single cell and is the cell's primary information storage medium. DNA stores all an organism's genetic information, including the instructions it needs to grow, divide, and live. DNA is made up of four different building blocks called nucleotide bases: adenine (A), thymine (T), cytosine (C), and guanine (G). The genome is sequenced in vitro utilizing encoding strategies such as labelling one bond pair as 0 and the other as 1 to store digital information. In this study, the fractional differential order of double-chain DNA dynamical system was investigated, considering Atangana’s Conformable fractional derivative. The Conformable sub-equation method was applied to the system.  The analysis resulted in some interesting new exact solutions of the model. One-soliton kink solution, multiple-soliton solutions, and periodic-wave solutions are the three broad categories that may be used to describe the results. In order to better understanding the solutions found, we have visually investigated a few of them. Both solitary and anti-solitary waves of the DNA strands are seen, attesting to the nonlinear dynamics of the system. The gathered data might be used to conduct application evaluations and draw further scientific findings.

In the present article, we utilize the Conformable Double Laplace Transform method (CDLTM) to get the exact solutions of a wide class of Conformable fractional di, er-ential in mathematical physics. The results obtained show that the proposed method is e, cient, reliable and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.

In this study‎, ‎we explore soliton solutions for the Conformable time-fractional Boussinesq equation utilizing the three-wave method‎. ‎To validate the precision of our findings‎, ‎we discuss specific special cases by adjusting certain potential parameters and also present the graphical representations of our results‎. ‎The results achieved in this research align closely with those from previous studies‎, ‎demonstrating enhanced accuracy and simplicity‎. ‎Given the extensive applications of this equation in particle physics‎, ‎understanding its dynamics is crucial‎. ‎Consequently‎, ‎employing methods that encompass a broad spectrum of solutions is imperative‎. ‎The versatility of this method in yielding diverse solutions is evident in the results we have obtained‎. ‎The solutions derived in this paper are novel and offer greater precision compared to previous works‎.

We expand a new generalization of the two-dimensional differential trans-form method. The new generalization is based on the two-dimensional differ-ential transform method, fractional power series expansions, and Conformable fractional derivative. We use the new method for solving a nonlinear con-formable fractional partial differential equation and a system of Conformable fractional partial differential equation. Finally, numerical examples are pre-sented to illustrate the preciseness and effectiveness of the new technique.

This paper deals with the solution of a class of Volterra integral equations in the sense of the Conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving Conformable fractional Volterra integral equations.

In this study, the Fan sub-equation method is applied to study exact solutions of complexnonlinear time fractional Maccari system. Using this method, we obtain various solutions which thoseare useful t to understand better the concepts of the complicated physical phenomena. This method isstraightforward, and it can be used to many nonlinear equations.

This paper presents a new sub-equation method based on an auxiliary equation which is implemented via the well-known generalized Kudryashov method, to construct new traveling waves to the Telegraph equation with time and space Conformable derivatives. To illustrate its effectiveness, it was tested for seeking traveling wave solutions to the (1+1)-Telegraph equation with space-time Conformable derivatives. With the help of Maple Software we derive some new solitary waves solutions. It can be concluded that the proposed method is an accurate tool for solving several kind of nonlinear evolution equations.

During the past years, a wide range of distinct approaches has been exerted to solve the nonlinear fractional differential equations (NLFDEs). In this paper, the invariant subspace method (ISM) in conjunction with the fractional Sumudu’ s transform (FST) in the Conformable context is formally adopted to deal with a nonlinear Conformable time-fractional dispersive equation of the fifth-order. As an outcome, a new exact solution of the model is procured, corroborating the exceptional performance of the hybrid scheme.