Quote:
Originally Posted by topsquark  So I'm working on this Physics problem, which (alas!) turns into an unsolvable Math problem. I have the equation (as a function of t): 
where a is presumed known.
The question boils down to how to find reasonable values for a' and b' that best model the above equation. (I can't simply do a Taylor Expansion on the left hand side because T could be large.) Now, being the Physicist I am I would know how find an estimate if the equation were 
The "best fit" in this case would be to find the average of the exponential function as t goes from 0 to T.
Is there a way to think of this equation as a "best fit" problem by using the exponential function as if it were "data?"
-Dan |
There are at least two answers to this (assuming a is a known numeric value)
1. Set up a model in Excel and use the solver to minimise:
![\int_{t=0}^T [e^{at}-(a't+b')]^2 dt](http://www.mathhelpforum.com/math-help/latex2/img/dc905dc70fc7864d4bdbcd9d53bc47f6-1.gif "\int_{t=0}^T [e^{at}-(a't+b')]^2 dt")
of course this will be discretised so you will have to find
that minimises:
![\sum_{t_i} [e^{at_i}-(a't_i+b')]^2](http://www.mathhelpforum.com/math-help/latex2/img/7b2992bc6c0f6c3974acae9d2d231b65-1.gif "\sum_{t_i} [e^{at_i}-(a't_i+b')]^2")
2. Expand
in orthogonal polynomials on
stopping at the linear term.
CB