Thanks everyone for the informative reply. Very appreciated!!!

If you're looking for something maybe more efficient and more clever, you can turn R(x)= ? P(x,t) dt from an integral equation into a differential equation.

Now that it's a differential equation you can use NDSolve or ParametricNDSolve:

https://reference.wolfram.com/language/ref/NDSolve.html

https://reference.wolfram.com/language/ref/ParametricNDSolve.html

Use NIntegrate to compute r[x] (here I use Mathematica notation and avoid capitalizing user-defined function names). Compute the integral of r[x]*f[x] also with NIntegrate (possibly it should just be done as a double integral). At that point you are pretty much done.

I should remark that the subject header seems to be incorrect; I have not found a specific example anywhere in this.

Hi Stven,

you say you need the relation between H(x,y) and y. But H does not depend on x, because x is only a dummy variable of integration. Then you simply have H(y) = h(y)*const, with: $$\mbox{const} := \int\mbox{d}x \; f(x) \int \mbox{d}t \; P(x,t)$$ Regards -- Henrik

Hi Stven,

The only thing I can think of is a rather brute-force and decidedly unelegant method: 1. Make a "sufficiently dense" (*) grid of points {x0, y0} in the (x,y}-plane. 2. Calculate your function H(x,y) on this grid. (3. This step is optional: Perhaps wrap an Interpolation command over the H(x,y} so as to get smooth connections between the grid points.) 4. Choose your fixed point x0 and plot H(x0, y) versus y.

This is how I would try it. Almost certainly there are more efficient methods, but this is simple and I don't see why it wouldn't work.

(*) Footnote re point 1: The "sufficiently dense" is the stumbling block of course. If you know next to nothing about H(x,y), you'll have to proceed by trial and error. The more you know about H, the smarter you can choose your grid: make the grid sparse in regions where H doesn't change much and dense in regions with dramatic gradients.

Hope this works for you.

René Samson