How to solve a time-varying polynomial with fractional exponents?

I am trying to solve an equation for R where all the parameters vary with time as shown below.

I have tried using a substitution of , but since the signals are time varying and of type double, the exponents might still be some fraction.

1. The above equation needs to be implemented in Simulink with R as Output while others being the inputs. Is there a way to solve the above equation of type

()

2. Since this is a very small part in the entire Simulink model, I am looking for something that can solve it without any explicit iterations or uses very little iterations.

3. I am hoping for a unique solution and would greatly appreciate if any insights can be made about uniqueness of the solution.

Answer 1

[John D'Errico]

First, why would you possibly home for a UNIQUE solution? Be serious. There will be multiple solutions, some of them probably complex. A polynomial problem will have multiple roots. Some of them will be complex, even most of them.

Next, why would you possibly hope for any analytical solution at all?

Sorry, but no. There is no analytical way to solve a totally general problem of the form

x^a - x^b = c

where a and b are known fractional and fully general constants.

We are not told what the values of f and gamma may be, only that they vary. Is gamma greater than 1? Less that 1 in general?

Can you solve it numerically? Well, of course. Use fzero as the simple solution.