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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 $t>0$?

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  • $\begingroup$ 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. $\endgroup$
    – Tucker
    Commented Aug 28, 2015 at 5:14
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    $\begingroup$ @Tucker So on the mathematical level there is no distinction. $\endgroup$
    – a06e
    Commented Mar 21, 2016 at 21:42
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    $\begingroup$ 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 $\endgroup$
    – Lutz Lehmann
    Commented Aug 12, 2022 at 6:28

1 Answer 1

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The names themselves telling the meaning. Boundary conditions are those which depends on space while initial conditions are depends on time. So, in boundary condition space coordinate will be varied and in initial conditions time will be varied.

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