Heat Equation

The heat equation is one of the essential partial differential equations that explain the distribution of heat in a given temperature of heat in given area over time. For a function u of three spatial variables and the time variable t, the heat equation is

Where α is a positive constant and  or  indicates the Laplace operator. In the physical problem of temperature difference u, is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.

The probability theory and heat equation

The heat equation is of a basic significance in various scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is related with the study of Brownian motion through the Fokker-Planck equation. In business mathematics, it is used to solve the Black-Scholes partial differential equation. The diffusion equation, a more general version of the heat equation arises in relation with the study of chemical diffusion and other related processes.

The maximum principle

Consider if one has a function u that explains the temperature at a given location. This operational will change over time as heat spreads throughout space. The heat equation is used to decide the change in the function u over time. The image to the right is animated and explains the way heat changes in time along a metal bar. One of the interesting properties of the heat equation is the maximum concept which says that the maximum value of u is either earlier in time than the region of concern or on the edge of the region of concern. 

Parabolic partial differential equations

This significantly says that temperature comes either from source or from earlier in time because heat permeates but is not made from nothingness. This is a property of parabolic partial differential equations and is not difficult to prove mathematically. Another fascinating property is that even if u has a discontinuity at an initial time t=t0, the temperature becomes smooth as soon as t > t0. for example if a bar of metal has temperature 0 and another has temperature 100 and they are stuck together end to end, then very rapidly the temperature at the point of relation is 50 and the graph of the temperature is smoothly running from 0 to 100.  The heat equation is employed in probability and explains random walks. It is also applied in business mathematics for this cause.

Questions:

• What is heat equation?
• What is the role of heat equation in chemical engineering?