Available online at www.CivileJournal.org Civil Engineering Journal (E-ISSN: 2476-3055; ISSN: 2676-6957) Vol. 8, No. 07, July, 2022 1358 A Computational Approach to a Mathematical Model of Climate Change Using Heat Sources and Diffusion Muhammad Shoaib Arif 1* , Kamaleldin Abodayeh 1 , Yasir Nawaz 2 1 Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, Saudi Arabia. 2 Department of Mathematics, Air University, PAF Complex E-9, Islamabad 44000, Pakistan. Received 20 April 2022; Revised 09 June 2022; Accepted 21 June 2022; Published 01 July 2022 Abstract The present work aims to extend the climate change energy balance models using a heat source. An ordinary differential equations (ODEs) model is extended to a partial differential equations (PDEs) model using the effects of diffusion over the spatial variable. In addition, numerical schemes are presented using the Taylor series expansions. For the climate change model in the form of ODEs, a comparison of the presented scheme is made with the existing Trapezoidal method. It is found that the presented scheme converges faster than the existing scheme. Also, the proposed scheme provides fewer errors than the existing scheme. The PDEs model is also solved with the presented scheme, and the results are displayed in the form of different graphs. The impact of the climate feedback parameter, the heat uptake parameter of the deep ocean, and the heat source parameter on global mean surface temperature and deep ocean temperature is also portrayed. In addition, these recently developed techniques exhibit a high level of predictability. Keywords: Energy Balance Models; Heat Sources; Diffusion Effects; Numerical Scheme; Stability. 1. Introduction In the 21 st century, climate change is the most frequently faced global problem. The world's future is highly dependent upon the study of climatic changes and their consequences. For this purpose, global climate models are the best way to anticipate any change. Commonly, a climate model provides a mathematical representation of the atmosphere, oceans, land, and their physical, chemical, and biological principles. These principles provide the basis for deriving equations which are solved numerically over a grid using discrete steps in space and time. The time period could range from a few minutes to many years depending upon the requirement of the process under observation or capacity of computer programing and the choice of the numerical method. Stability and scalability are the two main difficulties that climate researchers come across frequently. Stability refers to the stability of the solution with respect to initial conditions. However, scalability defines the increased resolution of current models. Nowadays, models with the highest resolution (mesoscale models) with too coarse a numerical grid exhibit difficulty in representing small-scale processes like turbulence in air and ocean boundary layers, interaction with small-scale topography features, thunderstorms, and cloud microphysics processes, etc. Fine-tuned approximations that were closely related to physical accuracy and computational viability were preferred by researchers. Partial differential equations are the key equations used to build climate models. These are non-linear equations with the system, and their solutions are truly significant. One can gain the most general cases' existence, uniqueness, and * Corresponding author: marif@psu.edu.sa http://dx.doi.org/10.28991/CEJ-2022-08-07-04 © 2022 by the authors. Licensee C.E.J, Tehran, Iran. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).