In mathematics, the **Dirichlet (or first-type) boundary condition** is a type of boundary condition named after a German mathematician Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, the condition specifies the values in which the derivative of a solution is applied within the domain’s boundary.

This condition corresponds to a given **fixed surface temperature **in heat transfer problems. The **Dirichlet boundary condition** is closely approximated, for example, when the surface is in contact with a melting solid or a boiling liquid. In both cases, there is heat transfer at the surface, while the surface remains at the temperature of the phase change process.

## General Heat Conduction Equation

The **heat conduction equation** is a partial differential equation that describes** heat** distribution (or the** temperature field**) in a given body over time. Detailed knowledge of the temperature field is very important in thermal conduction through materials. Once this temperature distribution is known, the **conduction heat flux** at any point in the material or on its surface may be computed from Fourier’s law.

The heat equation is **derived** from **Fourier’s law** and **conservation of energy**. Fourier’s law states that the time **rate of heat transfer** through a material is **proportional to** the negative **gradient in the temperature** and the area at right angles to that gradient through which the heat flows.

A change in internal energy per unit volume in the material, ΔQ, is proportional to the change in temperature, Δu. That is:

**∆Q = ρ.c**_{p}**.∆T**

**General Form**

Using these two equations, we can derive the general heat conduction equation:

This equation is also known as the **Fourier-Biot equation** and provides the basic tool for heat conduction analysis. From its solution, we can obtain the temperature field as a function of time.

In words, the **heat conduction equation** states that:

*At any point in the medium the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume.*