Systems of Ordinary Differential Equations (revised 3/2/2009)

  1. by Dennis Darland dennis.darland@yahoo.com

    dennisdarland.com

    Comments welcome

  2. I have a revised version of MapleSode.rb It generates a Maple 12 program to nuumerically solve a system of ordinary differential equations. It does not have the error estimation of the APFP used by Sode.rb which I worked on before. Although Maple 12 has 'tolerances', with my brief attempt to use them, I was unable to make comparison operations. The main advantage of Maple 12 is that its arbitrary precision floating point is much faster than APFP. ALSO: I found a basic problem in solving systems of ODEs in Sode.rb (this did not affect a single ODE). I have made a change for this in MapleSode.rb . (And I have made some improvements to Sode.rb from what I have learned in working on MapleSode.rb - but the channges in MapleSode.rb have been so extensive (due to the Language change) that I have not had time to retest Sode.rb thoroughly!) There is sample generated Maple 12 code and results of its output at MapleSode.rb MapleSode itself may be downloaded anonymously at MapleSode03.tar.gz

  3. Sode.rb is a Ruby program to generate a Ruby program to solve a single (or a system of) ordinary differential equation(s). (I (9/18/2008) have discpovered serious problems with its handling of systems of equations but single equationns were OK as far as I know. I have made an attempt to incorporate the fix from MapleSode.rb above, but it needs more testing.) A long Taylor series method, pioneered by Prof. Y.F. Chang, who taught at the University of Nebraska in the late 1970's when I was a graduate student there, is used. The number of Taylor series terms can be specified in the problem file, though it is usually 30. This was originally written in Unicon, but Unicon, although having overloaded operators in its CVS, still doesn't have then released, which caused me to switch to Ruby. During the development, especially in the beginning, it took an EXTREME effort on my part to work out the details and I made many mistakes (I suffer from a mental illness.) The result from tests seem generally good now. The complex example does not work as well as I would want. I wonder whether it is becaise the Taylor series for exponential function decrease slowly and don't alternate in sign. Anyway I get a significant number of downloads but no comments, suggestions or questions. It may be some people were disappointed in early versions, as it was a STRUGGLE for me, and I probably released to early and often, (as I thought this was part of the idea of OPEN SOURCE) but that kept me going.

  4. rdoc for Sode.rb

  5. rdoc for diffeq.rb for sin

  6. Y. F. Chang's Draft Typescript(1978) on Taylor Series [HOW IT WORKS] Chang on ATS

  7. The Taylor series terms are used to calculate the values of the dependent variables, and also (optionally) the radius of convergence and order of any singularities. These can be used to reduce the size of the next increment. Increasing it in the middle results in large errors because ecace sucessive term is multipiied by increasing powers of the ratio of the new $h to the old $h. In the begining I start with a very small $h and increase it until the estimated trucation error is too large and back up or else reach a maximum.

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