Solving Differential Equations in R

R Journal

Volumn 2, Issue 2, 2010, Pages 5-15

  Soetaert, Karline ^a   Petzoldt, Thomas ^b   Setzer, R Woodrow ^c  

^a NETHERLANDS INSTITUTE OF ECOLOGY NIOO KNAW   (Netherlands)

^b DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

^c U S ENVIRONMENTAL PROTECTION AGENCY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80052656348     PISSN: None     EISSN: 20734859     Source Type: Journal    
DOI: 10.32614/rj-2010-013     Document Type: Article

References (42)

1. U. Ascher, R. Mattheij, and R. Russell. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Philadelphia, PA, 1995.
2. U. M. Ascher and L. R. Petzold. Computer Methods for Ordinary Differential Equations and Differential- Algebraic Equations. SIAM, Philadelphia, 1998.
3. G. Bader and U. Ascher. A new basis implementation for a mixed order boundary value ODE solver. SIAM J. Scient. Stat. Comput., 8:483-500, 1987.
4. D. Bates and M. Maechler. Matrix: A Matrix Package for R, 2008. R package version 0.999375-9.
5. P. Bogacki and L. Shampine. A 3(2) pair of Runge- Kutta formulas. Appl. Math. Lett., 2:1-9, 1989.
6. K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM Classics in Applied Mathematics, 1996.
7. P. N. Brown, G. D. Byrne, and A. C. Hindmarsh. VODE, a variable-coefficient ode solver. SIAM J. Sci. Stat. Comput., 10:1038-1051, 1989.
8. J. C. Butcher. The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta, and General Linear Methods. Wiley, Chichester, New York, 1987.
9. J. R. Cash. 35 Test Problems for Two Way Point Boundary Value Problems, 2009. URL http://www.ma.ic.ac.uk/~jcash/BVP_software/PROBLEMS.PDF.
10. J. R. Cash, and A. H. Karp. A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides.ACM Transactions on Mathematical Software, 16:201-222, 1990.
11. J. R. Cash, and F. Mazzia. A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. J. Comput. Appl. Math., 184: 362-381, 2005.
12. A. Couture-Beil, J. T. Schnute, and R. Haigh. PBSddesolve: Solver for Delay Differential Equations, 2010. R package version 1.08.11.
13. J. R. Dormand, and P. J. Prince. A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math., 6:19-26, 1980.
14. E. Hairer, and G. Wanner. Solving Ordinary Differential Equations II: Stiff, and Differential-Algebraic Problems. Second Revised Edition. Springer-Verlag, Heidelberg, 2010.
15. E. Hairer, S. P. Noørsett, and G.Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Second Revised Edition. Springer-Verlag, Heidelberg, 2009.
16. S. Hamdi, W. E. Schiesser, and G. W. Griffiths. Method of lines. Scholarpedia, 2(7):2859, 2007.
17. A. C. Hindmarsh. ODEPACK, a systematized collection of ODE solvers. In R. Stepleman, editor, Scientific Computing, Vol. 1 of IMACS Transactions on Scientific Computation, pages 55-64. IMACS / North- Holland, Amsterdam, 1983.
18. W. Hundsdorfer, and J. Verwer. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2003.
19. S. M. Iacus. sde: Simulation, and Inference for Stochastic Differential Equations, 2008. R package version 2.0.3. IEEE Standard 754. Ieee standard for floating-point arithmetic, Aug 2008.
20. A. A. King, E. L. Ionides, and C. M. Breto. pomp: Statistical Inference for Partially Observed Markov Processes, 2008. R package version 0.21-3.
21. R. J. LeVeque. Finite Difference Methods for Ordinary, and Partial Differential Equations, Steady State, and Time Dependent Problems. SIAM, 2007.
22. F. Mazzia, and C. Magherini. Test Set for Initial Value Problem Solvers, release 2.4. Department of Mathematics, University of Bari, Italy, 2008. URL http://pitagora.dm.uniba.it/~testset. Report 4/2008.
23. L. R. Petzold. Automatic selection of methods for solving stiff, and nonstiff systems of ordinary differential equations. SIAM J. Sci. Stat. Comput., 4: 136-148, 1983.
24. T. Petzoldt. R as a simulation platform in ecological modelling. R News, 3(3):8-16, 2003.
25. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 3rd edition, 2007.
26. P. J. Prince, and J. R. Dormand. High order embedded Runge-Kutta formulae. J. Comput. Appl. Math., 7:67-75, 1981.
27. R Development Core Team. R: A Language, and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2009. URL http://www.R-project.org. ISBN 3-900051-07-0. T. Radivoyevitch. Equilibrium model selection: dTTP induced R1 dimerization. BMC Systems Biology, 2:15, 2008.
28. H. H. Robertson. The solution of a set of reaction rate equations. In J. Walsh, editor, Numerical Analysis: An Introduction, pages 178-182.Academic Press, London, 1966.
29. W. E. Schiesser. The Numerical Method of Lines: Integration of Partial Differential Equations.Academic Press, San Diego, 1991.
30. R. W. Setzer. The odesolve Package: Solvers for Ordinary Differential Equations, 2001. R package version 0.1-1.
31. K. Soetaert. rootSolve: Nonlinear Root Finding, Equilibrium, and Steady-State Analysis of Ordinary Differential Equations, 2009. R package version 1.6.
32. K. Soetaert, and P. M. J. Herman. A Practical Guide to Ecological Modelling. Using R as a Simulation Platform. Springer, 2009. ISBN 978-1-4020-8623-6.
33. K. Soetaert, and F. Meysman. ReacTran: Reactive Transport Modelling in 1D, 2D, and 3D, 2010. R package version 1.2.
34. K. Soetaert, and T. Petzoldt. Inverse modelling, sensitivity, and Monte Carlo analysis in R using package FME. Journal of Statistical Software, 33(3):1-28, 2010. URL http://www.jstatsoft.org/v33/i03/.
35. K. Soetaert, J. R. Cash, and F. Mazzia. bvpSolve: Solvers for Boundary Value Problems of Ordinary Differential Equations, 2010a. R package version 1.2.
36. K. Soetaert, T. Petzoldt, and R.W. Setzer. Solving differential equations in R: Package deSolve. Journal of Statistical Software, 33(9):1-25, 2010b. ISSN 1548- 7660. URL http://www.jstatsoft.org/v33/i09.
37. K. Soetaert, T. Petzoldt, and R. W. Setzer. R Package deSolve: Writing Code in Compiled Languages, 2010c. deSolve vignette - R package version 1.8.
38. K. Soetaert, T. Petzoldt, and R. W. Setzer. R Package deSolve: Solving Initial Value Differential Equations, 2010d. deSolve vignette - R package version 1.8.
39. M. H. H. Stevens. A Primer of Ecology with R. Use R Series. Springer, 2009. ISBN: 978-0-387-89881-0.
40. G. Strang, and G. Fix. An Analysis of The Finite Element Method. Prentice Hall, 1973.
41. C. W. Tornøe, H. Agersø, E. N. Jonsson, H. Madsen, and H. A. Nielsen. Non-linear mixed-effects pharmacokinetic/pharmacodynamic modelling in nlme using differential equations. Computer Methods, and Programs in Biomedicine, 76:31-40, 2004.
42. B. van der Pol, and J. van der Mark. Frequency demultiplication. Nature, 120:363-364, 1927.