meaning of fundamental solution

Question

I would like to understand what is a mathematical, even physical meaning of fundamental solution,let us consider following problem from Wikipedia.

$Lf=sin(x)$

where $L$ is operator of second derivative, as I know first it solves differential equation,that involves delta function like

because integral of delta function is Heaviside function H, or

we have

first of all in Wikipedia there is written,that for convenience they took constant $C=-1/2$ does it matter what we took?for example if we take $C=0$ or $C=10$? and finally what fundamental solution is convolution of right hand side functions with solution of delta function, like this

so my questions is

1.should we see another function instead of delta function,for example Dirac comb or others?

2.what is physical meaning of this convolution?

as i know solution related to delta function is called generalized function,or functions which are not continuous and have not derivatives,maybe i am wrong,but is it related to to such situation,when we have random events?or in deterministic events?hanks in advance

Answer 1

A fundamental solution to a linear differential operator $L$ is a distribution $E$ such that $L(E) = \delta$. One point of introducing these is that

$$L(E*f) = L(E)*f = \delta * f = f$$

(where $*$ denotes convolution). This means that you can create solutions to $L(u) = f$ simply by convolving $f$ with $E$. There are some technical assumptions here, in order to guarantee that everything is well defined.

Note that the fundamental solution is not uniquely determined. If $E$ is a fundamental solution and $v$ is any solution to the homogeneous equation $L(v) = 0$, then $E+v$ is also a fundamental solution. In other words, in your question it doesn't matter which $C$ you choose, but $C = -\frac12$ gives a nice "symmetric" choice of $E$.

The "physical meaning" is more or less that $E$ is an impulse response to the system. The physical interpretation of $\delta$ is a unit impulse: think of a very narrow, very high voltage peak or an almost instantaneous mechanical hit, like a pool cue striking a ball or a bouncing perfectly elastical ball or something like that.