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 the modern days where global environmental challenges are an emerging issue. This study explored application of ordinary differential equations in modelling of climate changes focusing on temperature trends. It used Newton’s law of cooling and heating as foundational physical and principles. ODEs are powerful mathematical tools in climate science as they enable modelling of transient and long term temperature responses to natural and anthropogenic factors. The study developed and analyzed first order ordinary differential equations based on Newton’s law which states that the rate at which an object cools off and heats up is proportional to the difference between the temperature of the object and that of the environment. 𝑑𝑇 𝑑� = −��(𝑇 − 𝑇� ) Where: �𝑇 -Rate of heat change with respect to time. �� T-Temperature of the object, 𝑇� - Environmental temperature and 𝑘 −Object property like ability of surface of objects to conduct heat. The equations were extended and modified to accommodate complex climate systems inherent in global and regional climate processes. Runge-Kutta method of order 4 was used to solve the equations in this study. Real world data from sources that is, NASA, NOA and Mauna Loa was used to validate the model where the model demonstrated high accuracy in simulating local temperature trends with an average error below 0.5ᵒC and strong agreement between observed and simulated values. All numerical simulations and graphical outputs were done using PYTHON software due to it’s flexibility and open source nature while the results were presented using tables and graphs.