Conservation of Heat Energy for Derivation of One-Dimensional Heat Equation and its Analytical Solution

Abstract

This article deals with the derivation of the one-dimensional heat equation and explores its analytical solutions, specifically focusing on Dirichlet boundary value and initial value problems associated with this particular partial differential equation. The primary objective of this study is to investigate the solution for the one-dimensional heat equation, which serves as a prominent example of a partial differential equation. To achieve this, the method of separation of variables is utilized to obtain the analytical solution. The heat equation, renowned for its effectiveness in analyzing the dynamic flow of heat within solids, has stood the test of time as a powerful tool for several centuries. Lastly, we have taken a model example and solve it using the stated method and also mesh plot is displayed using Python software.