We wish to calculate the integral: $$I=\int_{a}^{b} f(s) ds$$ Suppose, also, that there exists an operator $J$ which takes $f(s)$ such that:
$\int_{a}^{b} J[f](s) ds=\int_{a}^{b} f(s) ds=I$ and can be iterated such that $\int_{a}^{b} J^{n}[f](s) ds=\int_{a}^{b} f(s) ds=I$.
The mean value of $f(s)$ in $s:(a,b)$ is $\mu$ and $J$ operates in such a way that $|J^{n+1}[f](s)-\mu|<|J^{n}[f](s)-\mu|$ and $\lim_{n\to \infty} |J^{n}[f](s)-\mu|=0$ for all $s$ within $(a,b)$.
What sort of thing might $J$ be?
My own attempts to solve this make use of the heat equation, but I don't understand how to bound the heat equation such that none of the Quantity (heat) escapes the interval of integration.
I consider the heat equation because of its basis in conservation: Energy is a conserved quantity, so while it may be transferred by physical processes, it will remain trapped within matter and seek an equilibrium. The heat equation being $Q_{t}=k Q_{xx}$, if $T(t)$ is the total energy of the system at time $t$, we know that $T_{t}(t)=0$, so: $\int_{a}^{b} Q(x,t)dx=T(t)=T(0)$.
My (poor) attempt would be formulated thus:
Given: $$g(u,v)=J^{v}[f](u)$$ $$\int_{a}^{b} g(u,v) du=\int_{a}^{b} g(v,0) ds=G(v)$$ and $$G_{v}(v)=0$$ we say that $$g_{v}(U,V)=k•g_{uu}(U,V)$$
I would attempt to continue by naive first-order maclaurin recurrence approximation in my attempt to evolve the system through - what is essentially - time, by iteration using small $dv$: $$g(U,(n+1)dv)=g(U,(n)dv)+k•dv•g_{uu}(U,(n)dv)$$
Would being the operative phrase as this is however clearly not a feasible solution due to the lack of the aforementioned boundaries (the system evolves free to lose or gain area to the surrounding space).
As an example, my failure would thus be a evolution operator such that: $$J[h]=h+dn•k•h_{uu}$$
In theory, after an arbitrary period of evolving the system, I would then take a couple values from the iterated scheme and use them to approximate the definite integral (as the function would fill in its troughs with its hills and work its way toward the average value) by multiplying by the interval length.
So my question is: What sort of thing might $J$ be, and how would I formulate the boundaries?
PS. (I initially thought I might model the function by evolving it through something related to the fluid mechanics of a two dimensional water tank which starts with uneven distribution which would then be righted by gravity and brought to the equilibrium state of mean water depth, but I couldn't find any equations to describe this process.)