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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 specifi ed 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?

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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.

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  • $\begingroup$ thanks! your interpretation also helped a lot! $\endgroup$ – seraphimk May 22 at 18:55
  • $\begingroup$ It was no problem. $\endgroup$ – user3417 May 22 at 21:10
  • $\begingroup$ Just to be sure, the last type is called both Neumann B.C and Robin B.C ?, can you please clarify $\endgroup$ – BAYMAX May 25 at 20:35
  • $\begingroup$ @BAYMAX I edited it. I believe it can be called both Mixed and Robin conditions but somewhere I've seen people say there is a slight difference between Mixed and Robin. $\endgroup$ – user3417 May 25 at 20:40
  • $\begingroup$ cool! here, it mentions the difference between the mixed and Robin Boundary Conditions. Mixed BC essentially on disjoint parts of the domain and Robin on the whole of the domain. $\endgroup$ – BAYMAX May 25 at 20:44
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The boundary conditions are found by the physical condition of the problem at the end points.

For example if the end points are moving according to certain rule involving velocity or if the temperature at end points are controlled according to some rules dictated by the problem.

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