Numerical methods for ordinary differential equations

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

Relatit page: Numerical methods for pairtial differential equations

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:

• Bulirsch-Stoer algorithm^[6]
• Euler's method (namit after Leonhard Euler)
  • Backward Euler method^[7]
  • Semi-implicit Euler method
  • Euler-Maruyama method^[8]
• Exponential integrator^[9][10]
• Symplectic integrator^{[11][12][13][14]}
• Taylor series method^[15][16]

Validatit numerics for ODEs

Relatit page: Validatit numerics

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.

ODEs an relatit topics studiet in this context

• Blow-up solutions^[32][33]
• Lorentz equation^{[34][35][36][37][38][39]}
• Rossler equation^[40]

Relatit saftware

• Chebfun^{[41][42][43][44]}
• COSY INFINITY Archived 2020-09-24 at the Wayback Machine^[45][46]
• INTLAB^[21][47] and kv^[21][48] - interval arithmetic libraries includin ODE solvers
• MATLAB^{[49][50][51][52]}
• NAG library^[53][54][55]
• Wolfram Mathematica^{[56][57][58][59]}

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.