optimizing the solutions of differential equations using the lie groups method

Lie groups have wide applications in applied mathematics, including differential equations. in this article, the solution of two-dimensional differential equationsby the lie groups method, which is known as the dso(n) method, is investigated.the retention of lie groups structure under discretization is often vital in the recoveryof qualitatively correct geometry and dynamics and in the minimization of numericalerror. in this method, the jordan dynamics of a nonlinear dynamic system x˙ = f(x, t)where x = ‖x‖ in a quasi-linear system is deduced as the generalized hamiltoniandynamics of x with a diagonally symmetric and skew-symmetric coeffcient matrix.with this method, to the exact solution x(t) = g(t)x(0), g(t) dso(n) for a smalltime step with t 6 h , where h is a small size. a numerical example is provided toverify the correctness and effciency of the dso(n) method. the lyapunov exponentis defined for this method and the stability conditions of the system are determinedbased on the sign of the lyapunov exponent. finally, by numerical simulation, theoptimization of differential equations solutions using the lie groups method is investigated and validated by comparison with the rk4 method.