Lie group and RK4 for solving nonlinear first order ODEs


  • Sami H. Altoum umm Alqura University-university College of Qunfoudah





Lie group, Symmetry group, Rung-Kutta method, Numerical solution.


This paper deals with a numerical comparison between Lie group method and RK4 for solving an nonlinear ordinary differential equation. The Lie group method will be introduced as a analytical method and then compared to RK4 as a numerical method. Some examples will be considered and the global error we be computed numerically.


G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, (1989).

N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley Sons, West Sussex, (1999).

John. Starrett, “Solving Differential Equations by Symmetry Groupsâ€, American Mathematical Monthly, Vol.114, No.5, (2007), pp.778-792.

P. Glendinning, Stability instability an introduction to the theory of nonlinear differential equations, Cambridge England New York Cambridge University Press, (1987).

G. Dahlquist and A. Bjorck, Numerical methods in scientific computing, Society for Industrial and Applied Mathematics (SIAM)Philadelphia, (2008).

Grillakis. M, Shatah.J, Strauss.W, “Stability theory of solitary waves in the presence of symmetryâ€, Journal of Functional Analysis, Vol.74, No.3, (1987), pp.160-197.

Th. Simos, T. J. Kalvouridis.G, “plication of high-order Runge-Kutta methods in the magnetic-binaryâ€, An International Journal of Astronomy, Astrophysics and Space Science, Vol.147, No.2, (1988), pp. 271-285.

C. C. Christara and K. R. Jackson, “Scientific Computing by Numerical Methodsâ€, Encyclopaedia of Applied Physics, Vol.17, No.4, (1996), pp. 1-79.

Uri M.Ascher and Linda R.Petzold, Computer methods for ordinary Differential Equations and differential algebraic equations SIAM, (1998).

E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P “Computer algebra solving of second order ODEs using symmetry methodsâ€, Computer Physics Communications, Vol.108, No.3, (1988), pp. 90-114.

C.W.Gear, Numerical Initial Value Problems in Ordinary Differential Equations Prentice-Hall Englewood-Cliffs, (1971).

P.Phohomsiri and F.E Udwadia “Acceleration of Runge-Kutta integration schemesâ€, Discrete Dynamics in Nature and Socity, Vol.2, No.5, (2004), pp. 307-314.

View Full Article: