Studying the impacts of M-fractional and beta derivatives on the nonlinear fractional model

The major goal of the current research is to investigate the effects of fractional parameters on the dynamic response of soliton waves of fractional non-linear density-dependent reaction diffusion equation. Two well-known integration methodologies: the advanced exp(-Θ (ξ )) -expansion method and the modified auxiliary equation method in the sense of beta derivative and M-fractional derivative have been implemented to achieve explicit solitonic solutions of the fractional non-linear density-dependent reaction diffusion equation that emerged in mathematical biology. The spatial dynamics of populations, chemical concentrations, or other quantities are commonly studied using this equation type in biology, ecology, and chemistry. Solitary wave solutions of the governing equation, representing the dynamics of waves, plays a vital rule in many branches of biology, ecology, and chemistry. The obtained solutions has been studied in the form of singular kink-type solitary wave and kink-wave solutions. The behavior of soliton wave solutions is also demonstrated via 2D and 3D graphs. As a result of the fractional effects, physical changes are observed. The acquired results manifest that the proposed methods are more convenient, adequate, powerful and efficacious than other direct analytical methods. The attained results might improve our understanding of how waves propagate and could benefit the fields of medicine and allied sciences.