# What is the physics with examples behind the boundary conditions for heat, wave and Laplace equations?

Mainly there are three types of boundary conditions, TYPE I, TYPE II, TYPE III.

1. TYPE I is also known as Dirichlet conditions, i.e, $u(0,t)=f(t),\,\,\mbox{and}\,\,u(L,t)=g(t)$
2. TYPE II is also known as Neumann conditions, i.e, $u_x(0,t)=f(t),\,\,\mbox{and}\,\,u_x(L,t)=g(t)$
3. TYPE III is also known as Robin conditions, i.e, $\alpha u(0,t)+\beta u_x(0,t)=f(t),\,\,\mbox{and}\,\,\alpha u(L,t)+\beta u_x(L,t)=g(t)$

In the case of heat equation $u(x,t)$ is the temperature distribution and for wave equation $u(x,t)$ is the displacement.

Now can some one help me to understand the physics behind these conditions with practical examples?

Various types of heat transfer are good examples. A time-dependent temperature field $T(x,y,z,t)$ satisfies $${\partial T \over\partial t}=\nabla \cdot (\alpha\nabla T),$$ where $\alpha=\lambda/(\rho C)$ is the thermal diffusion coefficient for thermal conductivity $\lambda$, specific heat $C$, and density $\rho$. The diffusion coefficient may vary over space. Or in one dimension, $${\partial T \over\partial t} = \frac{\partial}{\partial x}\left(\alpha\frac{\partial T}{\partial x}\right).$$ Note that setting the left-hand side to zero will turn it into a Laplace equation (steady-state temperature).

A type I (Dirichlet) boundary condition would be to set the temperature to a fixed value at the boundary. Example: an object immersed in an ice-water mixture.

A type II (Neumann) boundary condition would be to set the heat flux to a fixed value. Heat flux is $q''=-\lambda \partial T/\partial x$. For example: a resistive heating element that dumps a fixed wattage of power into the system.

A type III (Robin) boundary condition could be radiative cooling: the heat flux at the boundary is proportional to the temperature difference with the environment. Suppose that the environment has temperature $T_0$; the heat flux condition at the rightmost boundary is then $\lambda\partial T/\partial x=h(T_0-T)$, where $h$ is a proportionality constant.

For wave equations, consider a standing wave in a pipe, with the displacement of the air molecules as your $u$ parameter. The three types correspond to a closed pipe end ($u=0$), an open pipe end ($u_x=0$), and a partially open pipe end.

• In the Laplace equation context, how then you have to interpret these boundary conditions? – zhk May 28 '16 at 16:16
• The same, but for a steady-state temperature distribution. And I just noticed that I should have written a first time derivative; will fix it now. – Han-Kwang Nienhuys May 28 '16 at 17:15
• Sorry, I noticed and corrected another error; specific heat and density was missing in the heat equation. (It's because I'm used to looking at steady-state heat transfer, not time-dependent ones). – Han-Kwang Nienhuys May 28 '16 at 17:47