This work uses the modified Extended Direct Algebraic Method (mEDAM) with conformable derivatives to obtain accurate solutions for the diffusion-reaction equation with cubic nonlinearity and the nonlinear fractional generalised density-independent DR problem. We use fractional derivatives in the conformable sense to achieve precise polynomial-form solutions by converting the equations into autonomous two-dimensional plane systems. The technique solves nonlinear ordinary differential equations (ODEs) produced by a fractional transformation to produce travelling wave solutions. Through the visualisation of various solution profiles, as periodic, shock, kink, anti-kink, and soliton waves, numerical simulations offer important insights into the dynamics of nonlinear fractional partial differential equations. This work builds on earlier research and broadens our understanding of complex events.

The present research work presents the modified Extended Direct Algebraic Method (m-EDAM) to construct and analyze propagating soliton solutions for fractional Kolmogorov-Petrovskii-Piskunov equation (FKPPE) which incorporates Caputo’s fractional derivatives. The FKPPE has significance in various disciplines such as population growth, reaction-diffusion mechanisms, and mathematical biology. By leveraging the series form solution, the proposed m-EDAM determines plethora of travelling soliton solutions through the transformation of FKPPE into Nonlinear Ordinary Differential equation (NODE). These soliton solutions shed light on propagation processes in the framework of the FKPPE model. Our study also offers some graphical representations that facilitate the characterization and investigation of propagation processes of the obtained soliton solutions which include kink, shock soliton solutions. Our work advances our understanding of complicated phenomena across multiple academic disciplines by fusing insights from mathematical biology and reaction-diffusion mechanisms.

In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

Purpose: This manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approach: The generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.Findings: The results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implications: The work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implications: The generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implications: Public health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/value: The use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

The nonlinear wave behavior in the tropical and mid-latitude troposphere has been simulated using the space-time fractional Landau-Ginzburg-Higgs model. These waves are the consequence of interactions between equatorial and mid-latitude waves, fluid flow in dynamic systems, weak scattering, and extended linkages. The mEDAM method has been used to obtain new and extended closed-form solitary wave solutions of the previously published nonlinear fractional partial differential equation via the beta derivative. A wave transformation converts the fractional-order equation into an ordinary differential equation. Several soliton, single, kink, double, triple, anti-kink, and other soliton types are examples of known conventional wave shapes. The answers are displayed using the latest Python code, which enhances the usage of 2D and 3D plotlines, as well as contour plotlines, to emphasise the tangible utility of the solutions. The results of the study are clear, flexible, and easier to replicate.