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