I am having some trouble understanding where some linear boundary conditions are derived from
The following is an extract from my lecture notes on boundary value problems for second-order Linear ODE's
In this section we are going to consider the different situation when some conditions are specified at the endpoints, or boundaries, of an interval of the independent variable, that is, at $x = x_1$ and $x=x_2$ with $x_1 < x_2$. This problem is known as a $ \textbf{Boundary Value Problem} $ and the conditions are called boundary conditions. We are then interested in finding the solution $y(x)$ to the ODE (which we consider to be Linear) inside the interval $x_1 \le x \le x_2$. we will consider only linear boundary conditions, where the left-hand sides of the conditions are linear combinations of the function and its derivatives at the same point and the right hand sides are given by constants, for example
$$ y(x_1) = b_1, \ y(x_2)=b_2 \ \ or \ \ y'(x_1)=b_1, \ y'(x_2) = b_2$$
or more generally
$$ \tag{1} \alpha y'(x_1) + \beta y(x_1)=b_1, \gamma y'(x_2)+\delta y(x_2) =b_2,$$
where $\alpha , \beta , \gamma , \delta$ are given real constants such that $|\alpha |+ | \beta | > 0, | \gamma | + | \delta|>0. $
My question is this, where do the conditions $(1)$ derive from?
Best Answer
There is no derivation. Those are the three types of boundary conditions generally seen.
$$ y(x_{1}) = b_{1}, y(x_{2}) = b_{2} $$
is known as a Dirichlet Boundary Condition
$$ y^{'}(x_{1}) = b_{1}, y^{'}(x_{2}) = b_{2} $$
is known as a Neumann Boundary Condition.
$$ \alpha y'(x_1) + \beta y(x_1)=b_1, \gamma y'(x_2)+\delta y(x_2) =b_2,$$
The last type is called Robin Boundary Conditions.
There is an interpretation.
Dirichlet Conditions mean you are holding the boundaries at a specific temperature.
Neumann Conditions means that the boundaries are being given energy at a specific rate.
Robin is a linear combination of the above.