Definition:Boundary Condition

Definition

Let $\Phi$ be a differential equation to which a particular solution is to be found.

A boundary condition is an equation relating particular values of the dependent and independent variables which the particular solution to $\Phi$ must satisfy.

It is usual for such boundary conditions to correspond to the physical extremities of bodies, aspects of whose internal nature is being modelled by means of $\Phi$.

Examples

Arbitrary Example

Consider the differential equation:

$\dfrac {\d^2 y} {\d x^2} + 4 \dfrac {\d y} {\d x} = 0$

for $x \ge 0$.

From Solution to $y + 4 y' = 0$ this has a general solution:

$y = A + B e^{-4 x}$

Let the boundary conditions be:

\(\ds y\)

\(=\)

\(\ds 0\)

when $x = 0$

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds 1\)

when $x = 0$

Then substituting $x = 0$ into the general solution and its first derivative yields:

\(\ds A\)

\(=\)

\(\ds \dfrac 1 4\)

\(\ds B\)

\(=\)

\(\ds -\dfrac 1 4\)

and so the particular solution:

$4 y = 1 - e^{-4 x}$

Also see

• Definition:Initial Condition

• Results about boundary conditions can be found here.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): boundary conditions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): boundary conditions
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): boundary condition