Analytical and numerical solutions to heat equations

Abstract

In this research, we study the heat equation. This equation originates from a scenario where a thin rod of finite length receives applied heat, defined as a function, while maintaining controlled temperatures on each end of the rod. Heat transfer throughout the rod over time is then described as a partial differential equation (PDE); the applied heat function gives the initial condition, and boundary conditions are the rod's end temperatures. As heat distributes along the rod’s axis, this mathematical model represents temperature with respect to position and time. The exact explicit solution to the problem is obtained by separation of variables, an analytical method for solving PDEs. A numerical method is used to approximate the solution by virtual simulation. Computer programming enables us to model and visualize the problem, as well as obtain and interpret numerical data.

The significance of this model is it allows us to quantify and simulate the heat distribution across a material satisfying the initial and boundary conditions. This research is applicable to heat movement in and out of a room, particle diffusion, thermal diffusivity in polymers, and heat transfer in biological systems.