Mathematical Modelling of Temperature Trends In Response to Climate Change Using Newton’s Law of Cooling.

Abstract

Understanding the dynamics of temperature trends is crucial for accurate climate

change modelling, especially in modern times, where global environmental chal- lenges are emerging. This study explored the application of ordinary differential

equations in climate change modelling, focusing on temperature trends. It used Newton’s law of cooling and heating as foundational physical principles. ODEs are powerful mathematical tools in climate science as they enable modelling of

transient and long-term temperature responses to natural and anthropogenic fac- tors. The study developed and analyzed first-order ordinary differential equations

based on Newton’s law. dT

dt = −k(T −Ta)(0.1) where: dT

dt Rate of heat change with respect to time, T Temperature of the object, Ta Environmental temperature, and k Object property like the ability of the surface of an object to conduct heat. The equations were extended and modified to accommodate complex climate systems inherent in global and regional climate processes. The 4th-order Runge-Kutta method was used to solve the equations in this study. Real-world data from

NASA, NOAA, and Mauna Loa sources validated the model. The model demon- strated high accuracy in simulating local temperature trends with an average error

below 0.5°C and strong agreement between observed and simulated values. All numerical simulations and graphical outputs were done using PYTHON software while the results were presented using tables and graphs.