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In this study, new numerical solutions of the conformable fractional Navier-Stokes equation are obtained by the conformable q-Shehu homotopy analysis transform method. The fractional Navier-Stokes equation is an extension of the classical Navier-Stokes equations that incorporates fractional derivatives to account for complex flow behaviors such as memory effects and anomalous diffusion. It is particularly useful in describing fluid dynamics in non-Newtonian fluids, porous media, and other systems with intricate time or spatial dependencies. In addition, two and three-dimensional graphs of the obtained solutions were drawn. We also conduct an error analysis to evaluate the accuracy of the scheme. Computational simulations are performed to validate the accuracy of the upcoming method. This paper presents the conclusions derived from the numerical and graphical analysis.

In this paper, the novel numerical method is used to analyze the conformable fractional Cahn-Hilliard equation. The Cahn-Hilliard equation is a mathematical model employed as a crucial tool in mathematical physics, specifically for understanding phase separation phenomena such as spinodal decomposition in systems with multiple phases. This study investigates the convergence and error of the proposed future scheme. The proposed technique produces h-curves that show the series solution's convergence interval. To ascertain the efficacy and appropriateness of this technique, the error analysis has been conducted. Also, 2D and 3D graphs of the solutions were drawn. Additionally, the behavior of the graphs was commented. This technique has been shown to be simple, effective and fast.

In the paper, the traveling wave solutions of the conformable derivative Date–Jimbo–Kashiwara–Miwa equation were obtained by the modified exponential function method (MEFM). It has been seen that the wave solutions found are functions that have the feature of being periodic functions. The proper values for the parameters in the acquired wave solutions are then used to generate two contour and density graphs in three dimensions that simulate the solution functions.

The present study focuses on the acquisition of traveling wave solutions associated with the conformable time-fractional Zoomeron equation through the utilization of the modified exponential function method (MEFM). The solution functions derived from mathematical computations encompass hyperbolic, trigonometric, and rational functions. Various graphical representations, such as 2D, 3D, contour graphs, and density graphs, are utilized to visually depict the distinct features of the solution functions derived from the determination of suitable parameters.

In this article, the Time Fractional Clannish Random Walker’s Parabolic Equation traveling wave solutions,a non-linear partial differential equation, is analyzed using the modified exponential function method (MEFM) and the Generalized Kudryashov Method (GKM). In this way, the solution functions of the mathematical model were obtained through a mathematical program with the help of two effective methods. Two-dimensional, three-dimensional, contour graphics simulating the behavior of this non-linear mathematical model were drawn with the help of the program under appropriate parameters.

In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist of hyperbolic, trigonometric, rational trigonometric, rational hyperbolic, and rational function solutions, respectively. This shows that the proposed methods are competent and sufficient. In addition, it is aimed to better understand the physical properties by drawing two- and three-dimensional graphics of the exact solutions according to different parameter values. When these exact solutions obtained by two different methods are compared with the solutions attained by other methods, it can be said that these two methods are competent.

In this work, traveling wave solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing nonlinear partial differential equations, which are examples of mathematical modeling, are obtained and analyzed using the modified exponential function method. In order to facilitate the physical interpretation of the mathematical models represented by these equations, simulations of the behavior of the mathematical model as three-dimensional, contour, density and two-dimensional graphics are given using a package program with the help of appropriate parameters. It has been shown that the modified exponential function method effectively investigates the solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing equations.

In this study, the exact solutions of the Biswas-Arshed equation with the beta time derivative, which has an important role and physically means that it represents the pulse propagation in an optical fiber, nuclear, and particle physics, are obtained using the modified exponential function method. Exact solutions consisting of hyperbolic, trigonometric, rational trigonometric, and rational function solutions demonstrate the competence and relevance of the proposed method. In addition, the physical properties of the obtained exact solutions are shown by making graphical representations according to different parameter values. It is seen that the method used is an effective technique, since these solution functions obtained with all these cases have periodic function properties.

In this article, the new function method is used to obtain the wave solutions of the nonlinear Klein-Gordon equation. Since the Klein-Gordon equation is a nonlinear partial differential equation containing exponential functions, it was decided to apply the new function method, which was defined on the assumption of a nonlinear auxiliary differential equation containing exponential functions. Thus, it aims to reach wave solutions not found in the literature. The considered method can be easily applied to this type of nonlinear problem that is difficult to solve and gives us solutions. Here, two new exact solutions are obtained. Then two and three-dimensional density and contour graphs are drawn by selecting the appropriate parameters to analyze the physical behavior of these solutions. The Mathematica package program was effectively used in all calculations and graphic drawings.

In this study, the traveling wave solutions of a nonlinear partial differential equation (NPDE) called as Geophysics Korteweg-de Vries (GpKdV) equation are obtained using the modified exponential function method (MEFM). Coriolis effect is stated with the help of this model used in geophysics. The nonlinear model has a Coriolis coefficient representing this effect.