I wish to know the methods of solving a system of equations that require, and those which don't require computer algebra. Is there a method, which is both theoretically and computationally efficient?
Sonam, I suggest you take a look at delay differential equations (DDEs) in matlab, and/simulink, mathworks .... all these in matlab. I am confident you will get exactly what you are looking for. You can also just Google for example solutions to problems on DDEs using matlab, there are numerous. Good luck.
Jimmy: Thank you, for your answer. I had already used matlab (mathworks, simulink) while solving a DDE model on Enzyme-Kinetics. Apart from that, I had used R and XPPAUT, before writing an amateur C program for validating my results. But I was not convinced with the inconsistency their solutions displayed, when I compared them with the analysis Variational Iteration Method had provided me with. So, my query is still unanswered, as I could not find any method, apart from the Runge-Kutta, that would give fairly good results. Thanks anyway for contributing. :)
As it was already suggested dde23 by Matlab is a good choice but it is Runge-Kutta based. There are however a few more that you may consider: I have had experience with ARCHI. It is a Runge-Kutta algorithm in FORTRAN language. There is also ddesolve. Depending the size of your system and the magnitude of the lag comparing to your simulation time, you could also consider successive approximation. It is however very time and resource consuming.
http://www.azimuthproject.org/azimuth/show/Numerical+methods+for+delay+differential+equations
Sonam i suggest R and deSolve package:
http://cran.r-project.org/web/packages/deSolve/vignettes/deSolve.pdf
Good luck
Zoltan, I already had done R and used deSolve. It gave me satisfactory results, though.
To test numerical solutions I recommend looking in a regime where centre manifold theory applies. The centre manifold of DDE systems may be reliably constructed using my web service at http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php
It applies for finite-D DDE systems near bifurcations of any co-dimension.
I also suggest you to check the :
DDE-BIFTOOL for bifurcation analysis of the equilibrium of the periodic solutions
http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml
http://arxiv.org/abs/1406.7144
or Knut (PDDE-CONT)
http://www.swmath.org/software/8420
Dear Sonam,
I would recommend the book
Springer Series in Computational Mathematics
Volume 8 1993
Solving Ordinary Differential Equations I
Nonstiff Problems
Authors:
Ernst Hairer,
Gerhard Wanner,
Syvert P. Nørsett
ISBN: 978-3-540-56670-0 (Print) 978-3-540-78862-1 (Online)
See also:
http://link.springer.com/book/10.1007%2F978-3-540-78862-1
I also have written two FORTRAN RK-codes which however work only for one constant delay; see the paper of Oberle/Pesch:
,,Numerical Treatment of Delay Differential Equations by Hermite Interpolation``,
Numer. Math. 37 (1981), 235-255. (DOI 10.1007/BF01398255)
The codes are from 1981! Unfortunately I do not have them anymore.
Please note that there will be not THE best Code for general Delay Diff. Eqs., but only the best one for your problem. The choice depends of the type of DDE. I am sure you will find an appropriate hint in the above mentioned book.
Best regards,
Hans Josef
I would suggest taking a look at the dde capabilities of Mathematica:
http://www.wolfram.com/products/mathematica/newin7/content/DelayDifferentialEquations/
http://reference.wolfram.com/language/tutorial/NDSolveDelayDifferentialEquations.html
http://reference.wolfram.com/language/howto/SolveDelayDifferentialEquations.html
http://www.wolfram.com/mathematica/new-in-10/enhanced-calculus-and-differential-equations/find-symbolic-solutions-for-delay-differential-equ.html
Best regards,
Mats
"to solve" could mean algebraic, numerical, or approximate. If you want to construct approximate emergent models near bifurcations (pitchfork, Hopf, higher order), you can use my web service that constructs a center manifold model of ODEs or DDEs for systems a user enters. My web page gives five examples you can try including a Double Hopf bifurcation in a delay DE.
Dear Sonam ,
may this tutorial help you
http://www.runet.edu/~thompson/webddes/tutorial.pdf
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