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 equations by 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 recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. 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 Hamiltonian dynamics 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 small time step with t 6 h , where h is a small size. A numerical example is provided to verify the correctness and effciency of the DSO(n) method. The lyapunov exponent is defined for this method and the stability conditions of the system are determined based on the sign of the lyapunov exponent. Finally, by numerical simulation, the optimization of differential equations solutions using the Lie groups method is investigated and validated by comparison with the RK4 method.