If we have a differential equation on [a, b], I know that a periodic boundary condition is written like $$f(a)=f(b)$$ in my book.
I am confused why I am being told in classes that $$f(a)=f(b)=0$$ is NOT a periodic boundary condition. I was hoping someone could enlighten me on this.
Thank you.
If we have the zero boundary conditions:
$$f(a)=f(b)=0$$
We are restricting ourselves only to the functions that are zero at those two points.
On the other hand, for the periodic boundary conditions, we have:
$$f(a)=f(b)=C$$
Where $C$ is some arbitrary (and usually unspecified) value. We could definitely set $C=0$, however calling this case periodic would be confusing.
The same way calling the function $y(x)= a$ a linear function would be confusing, just because we can set $b=0$ in the expression for the linear function $y(x)= a+bx$.
Usually, one calls the b.c. periodic when we can't say what value $C$ takes. We only know that $f(a)=f(b)$, that's it. In other words, we know less information about the function than in the case of zero boundary conditions.