I would like to implement fractional derivative in my simulation model. Most of the literature I find, discuss techniques of evaluating the fractional derivative using constant time-steps. Are there any methods which consider unequal time-steps and evaluate the fractional derivative because the solver of my simulation software does not guarantee a constant time-step.
I understand what a fractional derivative does basically. I am in search of a discrete approximation or any other numerical method to evaluate the fractional derivative of a function (say $x$) with respect to the simulation time. As far as I know, I shall consider the values of $x$ at various time steps (till the current simulation time is reached) and then evaluate the fractional derivative. The problem I have is that there are methods (like Grunwald Coefficients, etc.) which consider equal time step increments. But my solver does not have equal timesteps; instead, it selects the timestep based on the convergence of the solution. Let us assume a simulation time of 5 seconds with 500 number of timesteps. The methods I went through so far assume a time step of (5/500), i.e. 0.01s. So there the simulation time is as follows 0.01, 0.02, 0.03...., 4.99, 5.00. But what if my solver simulation time is as follows 0.01, 0.015, 0.024,..., 4.50, 5.00 where the increment is not a constant value? I want to know if there are any methods to solve such a case.