# What is the difference between boundary conditions and initial conditions?

What is the difference between boundary conditions and initial conditions?

I have two condition. The first is a boundary condition

$$\begin{equation} \theta (\mathbf{x},t)=k(\mathbf{x},t),\hspace{0.2cm}\mathbf{x} \in A, t>0 \end{equation}$$

And the second is the initial condition $$\begin{equation} \theta (\mathbf{x},0)=h(\mathbf{x}),\hspace{0.2cm}\mathbf{x} \in B, t=0 \end{equation}$$.

Why $$t=0$$ is not taken in boundary condition and in initial condition why not $$t>0$$?

• The boundary condition specifies the value that a solution must take in some region of space and is independent of time. The initial condition is a condition that a solution must have at only on instant of time. Aug 28, 2015 at 5:14
• @Tucker So on the mathematical level there is no distinction.
– a06e
Mar 21, 2016 at 21:42
• The question should specify for clarity that set $A$ is actually the boundary of some domain, while set $B$ is a domain, or the space part of the domain of the PDE. // Note that the use of "initial" and especially "boundary" is slightly different for ODE Aug 12, 2022 at 6:28