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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?

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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.

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  • $\begingroup$ In the Laplace equation context, how then you have to interpret these boundary conditions? $\endgroup$ – zhk May 28 '16 at 16:16
  • $\begingroup$ 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. $\endgroup$ – Han-Kwang Nienhuys May 28 '16 at 17:15
  • $\begingroup$ 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). $\endgroup$ – Han-Kwang Nienhuys May 28 '16 at 17:47

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