Heat Equation

Equation

• Let $\mathbf{B}\subset {\mathbb{R}}^2$ denote a physical body in 2D space
• Suppose that $T(x,y,t)$ is the temperature at time $t$ of point $(x,y)\in \mathbf{B}$
• The temperature must satisfy the heat equation:

$$k(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})=\frac{\partial T}{\partial t}$$

Where:

• $k$ is a constant describing the heat in body $\mathbf{B}$

Solutions

A solution to the heat equation satisfies: $T(x,y,t)=e^{-kt}(\cos (x)+\cos (y))$