Can someone provided some insight on what is meant by "weighted average"?
Allow me to give some exposition.
So given the heat equation on the whole line
$$\begin{cases} u_{t}-ku_{xx}=0\quad\quad -\infty <x<\infty ,\;t>0\\ u(x,0)=\phi (x) \end{cases} $$
we know its solution is given by
$$u(x,t)=\int_{-\infty}^{\infty}S(x-y,\;t)\phi(y)\;dy,$$
where $S(x,t)$ is the heat kernel
$$S(x,t)=\frac{1}{\sqrt{4\pi kt}}\;e^{-x^2/4kt}\quad t>0.$$
The heat kernel, as we know it, represents the evolution of temperature for some region in space. In this case, for a 1-D infinite rod.
Now, in Walter A. Strauss' An Introduction to Partial Differential Equations, Strauss claims that the solution $u(x,t)$ is "a kind of weighted average of the initial values around the point $x$."
He later justifies this notion by writting the solution as follows
$$\int_{-\infty}^{\infty}S(x-y,\;t)\phi(y)\;dy\simeq \sum_t S(x-y_i\;t)\phi(y_i)\Delta y_i $$
and suggests that the solutions $S(x-y_i)$ are weighted by $\phi(y_i)$ .
My interpretation of the solution (in the context of 1-D heat flow in a rod) is that once time starts, the initial temperature distribution $\phi(x)$ spreads throughout the rod and as time goes on, the temperature value at a fixed point $x$ is given/determined by the average value the temperature of nearby points.
Is my intuition on track?
Also, what exactly does Strauss mean when he says $u$ is a "weighted average"?
Any input is much appreciated.
The literal interpretation of this formula is as follows: The function $S(x-y_1)$ is the normal distribution shifted from $x=0$ to a specific $x$ value $= y_1$. All values of this shifted function along the $x$-axis is then multiplied by all the corresponding values along the $x$-axis of the (unshifted) initial condition function. This number is multiplied by delta $y$. All these numbers are added up. As the area below the normal function is $1$, the resulting sum becomes a weighed average, i.e. the initial condition is weighed by the corresponding values of the normal distribution located at the specific position $x=y_1$. To get the value at another $x$ position, you have to shift the normal function to that new $x$-position and repeat the calculation. The procedure may be described as finding a moving average or as a convolution.