<p>Many scientific fields have emphasized studying the dynamic behavior associated with environmental occurrences. Warming of the planet is one such occurrence. The two primary drivers of global warming negatively impacting our ecosystem are temperature and excess greenhouse gases. In light of the importance of this climatic event, we have considered the three climatic variables in this study: temperature, greenhouse gas concentrations, and permafrost thawing in the form of a fractional-order atmospheric model. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The atmospheric model is transformed into an algebraic equation system using the collocation approach and the fractional derivative operational matrices. The Newton–Raphson method of solving these algebraic equations entails inserting the estimated values of the generated unknown coefficients. The present method, ND solver, and RK methods are compared numerically. Furthermore, we demonstrate the method’s effectiveness in various scenarios by running simulations with fractional orders and parameter values. The results confirm the method’s capacity to produce accurate results in many contexts. Tables and graphs show how well the developed strategy performed over time and how effective it was. These results help us analyze the variation of three dependent variables by varying parameter values. The Hermite wavelet collocation method described here is accurate in terms of convergence and computational cost and robust when compared to previous methods in the literature. Mathematica is a mathematical software for numerical computations.</p>