1D Heat Equation

The 1D Heat Equation models the diffusion of heat along a one-dimensional medium over time. It explains how temperature changes are distributed due to thermal conductivity.

The 1D Heat Equation project focused on modeling and understanding heat diffusion along a one-dimensional medium, such as a rod. This equation describes how temperature changes over time due to thermal conductivity. The project involved solving the equation using numerical methods, including explicit and implicit finite difference schemes. Various scenarios were explored, such as uniform initial temperature distributions, localized heat sources, and different boundary conditions like fixed temperatures (Dirichlet), insulated ends (Neumann), and mixed conditions. Using MATLAB, the temperature evolution was visualized through dynamic plots and heatmaps, highlighting the effects of thermal diffusivity alpha on the rate of heat flow. For instance, materials with higher α\alphaα values, such as metals, exhibited faster heat diffusion, while those with lower values, like glass, showed slower changes. The project also investigated custom cases, such as the propagation of a heat pulse and multi-material rods with varying properties. These simulations validated theoretical predictions, demonstrating how heat spreads and eventually reaches thermal equilibrium. The project showcased the practical applications of the 1D Heat Equation in engineering, environmental science, and material studies, providing an insightful blend of mathematics, computational modeling, and real-world relevance.