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

enter image description here

because integral of delta function is Heaviside function H, or

enter image description here

we have

enter image description here

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

enter image description here

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


1 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.

  • $\begingroup$ in other words,it describes situation how impulse response affects process determined by some function? $\endgroup$ Sep 19, 2013 at 7:16
  • $\begingroup$ @mrf Re. "The "physical meaning" is more or less that E is an impulse response to the system" i think this is not right. Your equation $L(E)=\delta$ shows that E is the input to the operator that would produce an output equal to $\delta$. The impulse response I would be $L(\delta)=I$. $\endgroup$
    – user45664
    Dec 31, 2019 at 18:35
  • $\begingroup$ @user45664 Actually, in the physical interpretation, the right hand side of the equation can be considered as the input to the dynamical system, and the solution is the output or response, so $E$ is the response to the input $\delta$, i.e. the "impulse response". $\endgroup$
    – mrf
    Jan 1, 2020 at 21:06

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