Numerical methods for ordinary differential equations

**Numerical methods for ordinary differential equations** are computational schemes tae obtain approximate solutions o ordinary differential equations (ODEs).

## Background
Syne odes appearit i science, many mathematicians have studiit hou tae solve thaim. However, only few o thaim can be mathematically solvit. This is why numerical methods are needit. Ane o the most famous methods are the Runge-Kutta methods, but it disnae wirk for some ODEs (especially nonlinear ODEs). This is hou new ode solvers are developit. The followin list includes frequently uised methods:

* Bulirsch-Stoer algorithm
* Euler's method (namit after Leonhard Euler)
** Backward Euler method
** Semi-implicit Euler method
** Euler-Maruyama method
* Exponential integrator
* Symplectic integrator
* Taylor series method

## Validatit numerics for ODEs

Nae anely approximate solvers, but the study tae "verify the existence o solution bi computers" is also active. This study is needit acause numerically obtaint solutions cud be phantom solutions (fake solutions). This kynd o incident is awreidy reportit. The popular methods are basit on the shootin method or spectral methods. The day, European resairch teams an Japanese experts ar wirkin on this topic.
### ODEs an relatit topics studiet in this context

* Blow-up solutions
* Lorentz equation
* Rossler equation
### Relatit saftware

*Chebfun
*COSY INFINITY 
* INTLAB and kv - interval arithmetic libraries includin ODE solvers
* MATLAB
* NAG library
* Wolfram Mathematica

## Forder readin

* Mitsui, T., & Shinohara, Y. (1995). Numerical analysis of ordinary differential equations and its applications. World Scientific.
* Iserles, A. (2009). A first course in the numerical analysis of differential equations. Cambridge University Press.
* Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag.
* Wanner, G. & Hairer, E. (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.). Springer Berlin Heidelberg.
* Butcher, John C. (2008), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons.
* John D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
* Deuflhard, P., & Bornemann, F. (2012). Scientific computing with ordinary differential equations. Springer Science & Business Media.
* Shampine, L. F. (2018). Numerical solution of ordinary differential equations. Routledge.
* Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press.

## References

## Fremmit airtins
* Joseph W. Rudmin, *[http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf Application of the Parker–Sochacki Method to Celestial Mechanics] *, 1998.
* Dominique Tournès, *[https://web.archive.org/web/20130413090625/http://www.reunion.iufm.fr/dep/mathematiques/calculsavant/Equipe/tournes.html L'intégration approchée des équations différentielles ordinaires (1671-1914)]*, thèse de doctorat de l'université Paris 7 - Denis Diderot, juin 1996. Réimp. Villeneuve d'Ascq : Presses universitaires du Septentrion, 1997, 468 p.
*INTLAB 
*Verified ODE (IVP) Solver