This work addressed one of the most essential evolutionary equations that was widely used in describing various nonlinear wave propagation and dispersive phenomena in various scientific and engineering applications, which was called the nonlinear fractional Fornberg-Whitham (FFW). Due to the importance of this equation, we examined it by employing two highly effective techniques: the residual power series method (RPSM) and the new iterative approach (NIM), both distinguished by their efficacy in solving more complicated nonlinear fractional evolutionary equations. Moreover, we integrated the Elzaki transform with both approaches to create Elzaki RPSM (ERPSM) and Elzaki NIM (ENIM) to ease the calculations. The ERPSM effectively combined the power series approach with residual error analysis to generate highly accurate series solutions, while ENIM provided alternative frameworks for handling nonlinearities and achieving rapid convergence. Comparative studies of the obtained solutions highlighted these methods' efficiency, accuracy, and reliability in solving fractional-order differential equations. The results underscored the potential of these analytical techniques for modeling and solving complex fractional wave equations, contributing to the advancement of mathematical physics and computational fluid dynamics.

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water.

This paper solves an example of a time-space fractional Kuramoto-Sivashinsky (KS) equation using the integrated Bäcklund transformation and the Riccati-Bernoulli sub-ODE method. A specific version of the KS equation with power nonlinearity of a given degree is examined. Using symbolic computation, we find new analytical solutions to the current problem for modeling many nonlinear phenomena that are described by this equation, like how the flame front moves back and forth, how fluids move down a vertical wall, or how chemical reactions happen in a uniform medium while they oscillate uniformly across space. In the field of mathematical physics, the Riccati-Bernoulli sub-ODE approach is shown to be a valuable tool for producing a variety of single solutions.

In this research work, we use the generalized Bernoulli equation method (gBEM) to investigate the chaotic nature of solitons in complex structured conformable coupled Maccari system (CCMS) which is a fractional generalization of a nonlinear model coupled Maccari system (CMS) initially created to simulate hydraulic systems. This key model has numerous applications in several disciplines such as hydrodynamics, optics, quantum mechanics and plasma physics. The proposed gBEM converts the CCMS into a set of nonlinear ordinary differential equations (NODEs) to construct fresh plethora of soliton solutions in the form of rational, trigonometric, hyperbolic, and exponential functions. To comprehend the dynamics of acquired solitons in CCMS, a series of 3D and counter plots are used which graphically illustrate and reveal two types of chaotic perturbations, namely axial and periodic perturbations in the acquired solitons. Furthermore, the efficiency and adaptability of our method in handling a range of nonlinear models in mathematical science and engineering are confirmed by our computational work.

This paper presents a new scientific method to obtain the precise soliton solutions of the fractional conformable Kairat-Ⅱ (K-IIE) and Kairat-X (K-XE) equations using the Riccati-Bernoulli sub-ODE technique and the Bäcklund transformation. These methods seem most effective compared to the previously used methods as they demonstrate novelty in intensity and potential application. The solutions attained in form of trigonometric, hyperbolic, and rational forms can be used in areas of nonlinear optics, ferromagnetic dynamics, photonic crystals, and optical fibers theory. The most important findings are displayed in simple $2D$ graphs in order to illustrate the nature of these solutions. The use of the flexible fractional derivatives shows that the models are integrated and provides opportunities for studying differential geometry and curve equivalence. Similarly, the ease of the methods underscores applications in other nonlinear partial differential equations, affirming their versatility and importance for the upper level courses.

The integrable Kuralay-Ⅱ system (K-IIS) plays a significant role in discovering unique complex nonlinear wave phenomena that are particularly useful in optics. This system enhances our understanding of the intricate dynamics involved in wave interactions, solitons, and nonlinear effects in optical phenomena. Using the Riccati modified extended simple equation method (RMESEM), the primary objective of this research project was to analytically find and analyze a wide range of new soliton solutions, particularly fractal soliton solutions, in trigonometric, exponential, rational, hyperbolic, and rational-hyperbolic expressions for K-IIS. Some of these solutions displayed a combination of contour, two-dimensional, and three-dimensional visualizations. This clearly demonstrates that the generated solitons solutions are fractals due to the instability produced by periodic-axial perturbation in complex solutions. In contrast, the genuine solutions, within the framework of K-IIS, take the form of hump solitons. This work demonstrates the adaptability of the K-IIS for studying intricate nonlinear phenomena in a wide range of scientific and practical disciplines. The results of this work will eventually significantly influence our comprehension and analysis of nonlinear wave dynamics in related physical systems.

In this work, the Extended Direct Algebraic Method (EDAM) is utilized to analyze and solve the fractional (2+1)-dimensional Conformable Broer-Kaup-Kupershmit System (CBKKS) and investigate different types of traveling wave solutions and study the soliton like-solutions. Using the suggested method, the fractional nonlinear partial differential equation (FNPDE) is primarily reduced to an integer-order nonlinear ordinary differential equation (NODE) under the traveling wave transformation, yielding an algebraic system of nonlinear equations. The ensuing algebraic systems are then solved to construct some families of soliton-like solutions and many other physical solutions. Some derived solutions are numerically analyzed using suitable values for the related parameters. The discovered soliton solutions grasp vital importance in fluid mechanics as they offer significant insight into the nonlinear behavior of the targeted model, opening the way for a deeper comprehension of complex physical phenomena and offering valuable applications in the associated areas.

We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.