The main target of this work is presenting two efficient accurate algorithms for solving numerically one of the most important models in physics and engineering mathematics, Fisher–Kolmogorov–Petrovsky–Piskunov’s equation (Fisher‐KPP) with fractional order, where the derivative operator is defined and studied by the fractional derivative in the sense of Liouville–Caputo (LC). There are two main processes; in the first one, we use the compact finite difference technique (CFDT) to discretize the derivative operator and generate a semidiscrete time derivative and then implement the Vieta–Lucas spectral collocation method (VLSCM) to discretize the spatial fractional derivative. The presented approach helps us to transform the studied problem into a simple system of algebraic equations that can be easily resolved. Some theoretical studies are provided with their evidence to analyze the convergence and stability analysis of the presented algorithm. To test the accuracy and applicability of our presented algorithm a numerical simulation is given.

The complex Hirota-dynamical Model (HDM) finds multifarious applications in fields such as plasma physics, fusion energy exploration, astrophysical investigations, and space studies. This study utilizes several soliton-type solutions to HDM via the modified simple equation and generalized and modified Kudryashov approaches. Modulation instability (MI) analysis is investigated. We also offer visual representations for the HDM.

This research uses three analytical processes: the novel Kudryshov procedure, the direct algebraic approach, and the modified Kudryshov technique to acquire exact and analytical outcomes for the long–short wave interaction nonlinear problem. A variable transformation is applied to convert the system’s partial differential framework into an ordinary one. Thereafter, the governing model is analyzed using the methods mentioned above, which yield novel dynamic optical solitons. These solutions include bell-shaped solitons, multiple breathers with bright, dark, and dark–bright structures, as well as periodic breathers with bright, dark, and bright–dark forms. Demonstrated through stable solutions, these results validate the computational approach. Multistability and sensitivity are explored using employing planar dynamics. Various chaos-identifying tools such as bifurcation plots, return maps, Lyapunov exponents, and power spectrum plots are applied to test the chaotic analysis of the stated model. Moreover, quasi-periodic, periodic, and disordered patterns are derived from the presented equation. Additionally, graphical representations illustrate the dynamic aspects of the obtained solutions. Overall, the employed methodologies prove impactful, efficient, and relevant for tackling diverse nonlinear phenomena.

In this manuscript, the existence and uniqueness of solutions to equations associated with climate change are discussed. For this purpose, we utilize some results from the existing literature to investigate the behavior of these equations. Additionally, the role of fixed point theory in emphasizing the importance of proving the stability and consistency of the models is explored. Several definitions and results, such as the F-contraction, α -F-contraction, rational type (ψ,ϕ) -contraction, and Geraghty type contraction, are recalled from the existing literature to illustrate their theoretical foundations and practical applications.