- #1
FranzS
- 88
- 26
- TL;DR
- Mixed approximation vs. full approximation for a power series expansion: which works better?
Hi PF,
I'm trying to find an approximate solution of a differential equation that can't be solved in exact form.
The differential equation is of the form:
$$
f'(t)=g(f(t),t)
$$
I want to find the approximate solution in terms of a power series:
$$
f(t) \approx f(0) + f'(0) \cdot t + \frac{f''(0)}{2} \cdot t^2
$$
##f(0)## is known, as is everything else in order to find ##f'(0)## directly from the differential equation.
The problem arises when calculating ##f''(0)##, because ##g(f(t),t)## includes a square root term and its derivative will diverge for ##t=0##.
Now, I found a good approximation for the square root term that has no divergence problems when differentiated.
My question is: for the power series expansion of ##f(t)##, should I use the term ##f'(0)## that was calculated with the exact form of ##g(f(t),t)## even though the term ##f''(0)## comes from the approximation of ##g(f(t),t)## or should I use the latter to find a new ##f'(0)## as well? How would those two options differ numerically and conceptually?
Thanks.
I'm trying to find an approximate solution of a differential equation that can't be solved in exact form.
The differential equation is of the form:
$$
f'(t)=g(f(t),t)
$$
I want to find the approximate solution in terms of a power series:
$$
f(t) \approx f(0) + f'(0) \cdot t + \frac{f''(0)}{2} \cdot t^2
$$
##f(0)## is known, as is everything else in order to find ##f'(0)## directly from the differential equation.
The problem arises when calculating ##f''(0)##, because ##g(f(t),t)## includes a square root term and its derivative will diverge for ##t=0##.
Now, I found a good approximation for the square root term that has no divergence problems when differentiated.
My question is: for the power series expansion of ##f(t)##, should I use the term ##f'(0)## that was calculated with the exact form of ##g(f(t),t)## even though the term ##f''(0)## comes from the approximation of ##g(f(t),t)## or should I use the latter to find a new ##f'(0)## as well? How would those two options differ numerically and conceptually?
Thanks.