ANALYSIS OF DIFFERENTIAL EQUATIONS FOR MATHEMATICAL MODELING OF PHYSICAL PHENOMENA

Differential equations are an effective mathematical tool for the mathematical modelling and investigation of a wide range of physical events. In this study, differential equations' involvement in the process of mathematical modelling and comprehension of physical systems will be studied. We show the usefulness of differential equations in capturing and characterising complex behaviours and dynamics displayed by physical systems by analysing the fundamental ideas and methods used in differential equations. The essential ideas of differential equations, including their classification and characteristics, are presented in the study's opening section. We address in depth a number of differential equation forms that are frequently found in physics, such as ordinary differential equations (ODEs) and partial differential equations (PDEs). This study focuses on the mathematical simulation of heat conduction in the presence of heat sources, which has numerous technical applications, such as heat diffusion, urban air pollution, and chemical reactions. To comprehend heat-related physical phenomena, the research uses methods including polynomial approximation, variability analysis, and numerical approaches. The study suggests a mathematical framework for investigating the transmission of heat in a metal bar that has a heat source. The parameter variation method is initially used to acquire the precise solution of the mathematical model. This method makes it possible to comprehend the behaviour of the system under investigation completely.