# 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.[1][2][3][4] 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,[5] 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:

## 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.[17][18] The popular methods are basit on the shootin method or spectral methods.[19][20] The day, European resairch teams[21][22][23][24][25][26][27][28][29] an Japanese experts[30][31] ar wirkin on this topic.

### Relatit saftware

• 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.

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