The "quadrature method" that I demonstrated to the OP is quite flexible for a single nonlinear ODE. The Schaeffer-Pella-Tomlinson ODE that you are referring to can be readily solved by the quadrature method. It should be significantly more efficient (i.e. accuracy/speed trade-off) than numerical ODE solvers.
The only situation where the quadrature method will not work for a single ODE is when the dependent variable and time cannot be separated, such as, for example: dy/dt = a*y^b*(1 - y) + g(t) where g(t) is a non-autonomous forcing function, which can be any function of time such that the solution is bounded as t goes to infinity. Ravi. ------------------------------------------------------- Ravi Varadhan, Ph.D. Assistant Professor, Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University Ph. (410) 502-2619 email: rvarad...@jhmi.edu -----Original Message----- From: dave fournier [mailto:da...@otter-rsch.com] Sent: Friday, December 17, 2010 7:02 AM To: Ravi Varadhan Cc: r-help@r-project.org Subject: Re: [R] Solution to differential equation Ravi Varadhan wrote: Because the numerical solution is more flexible. In the example I linked to the population is being fished. This add an extra term which breaks your solution. I don't know where the OP is going with this question, but flexibility might be useful. Also I just like the idea of fitting models defined by DE's to data. > When you can obtain `exact' (but not closed-form) solution, why would you > want to use a numerical ODE solver, which has an approximation error of the > order O(dt) or O(dt^2), where `dt' is the time step? Furthermore, a > significant advantage of an exact solution is that you can compute the > solution at any given `t' in one shot, rather than having to march through > time from t=t0 to t=t. Numerical time-marching schemes make more sense for > systems of nonlinear ODEs. > > Ravi. > > ------------------------------------------------------- > Ravi Varadhan, Ph.D. > Assistant Professor, > Division of Geriatric Medicine and Gerontology School of Medicine Johns > Hopkins University > > Ph. (410) 502-2619 > email: rvarad...@jhmi.edu > > > -----Original Message----- > From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On > Behalf Of dave fournier > Sent: Friday, December 17, 2010 11:23 AM > To: r-help@r-project.org > Subject: Re: [R] Solution to differential equation > > > It is not very difficult to integrate this DE numerically. > For parameter estimation it is a good idea for > stability to use a semi-implicit formulation. The idea is > described here. > > http://otter-rsch.com/admodel/cc4.html > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > > > ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
