Thermal equation
Text written by Bojan Crnković, Faculty of Mathematics, University of Rijeka
The development of mechanics and thermodynamics started after Newton and Leibnitz developed infinitesimal calculus. This led to the use of models based on partial differential equations. Partial differential equations (PDEs) play a fundamental role in modeling and predicting various physical phenomena, including heat conduction and diffusion processes in nature. These equations describe how quantities such as temperature and concentration depend on independent variables such as time and spatial coordinates. In the context of heat conduction and diffusion, PDEs are important tools for understanding how heat energy or substances spread through different media. One of the most famous PDEs used in the modeling of heat conduction and diffusion is the heat equation, also known as the diffusion equation. It describes how the temperature distribution in the material changes over time. The equation can be written in this way:
\frac{\partial u(x,t)}{\partial t}=\alpha \Delta_x u(x,t), where u is the temperature, x is the spatial coordinate, t is the time and α is the thermal diffusivity of the material.
By solving the heat equation under appropriate boundary and initial conditions, scientists and engineers can predict how heat propagates in solids, liquids, and gases. This has numerous practical applications in engineering, physics, chemistry, biology… In addition, interdisciplinary collaboration between mathematicians, physicists, engineers, and biologists contributes to the development of innovative approaches and the application of PDE models to new areas. Thus, these models are used in computer graphics, space search planning, robot organization, division of labour and intelligent systems management. One of these applications is shown in the figure, which represents the use of the heat equation and the HEDAC algorithm to make robots paint a picture using multiple brushes at the same time. Using this algorithm, an entire scientific exhibition called “[ai] explore” was created, which you can see at: https://socri.uniri.hr/ai-explore/.