A real time Lie group differential algebraic equations method to solve the inverse nonlinear vibration problems
Desiring a real time recovery of an unknown external force in the nonlinear inverse vibration problem, we transform the nonlinear ordinary differential equation (ODE) of motion into a nonlinear parabolic type partial differential equation (PDE) which can raise the robustness against large noise. Then, we come to a heat source identification problem, of which the numerical method of lines is used to discretize the PDE into a system of differential algebraic equations (DAEs) Consequently, we can develop an implicit Lie group scheme and a Newton algorithm to stably solve the DAEs for finding the unknown force, damping function, or stiffness function, which is well recovered even under a large noise. Superficially, we seem to transform a simple ODE to a more complex PDE; however, the advantages gained in this transformation will be seen when we test some nonlinear inverse vibration problems with large time span and under large noise.