A Lie-group adaptive differential quadrature method to identify an unknown force in an Euler–Bernoulli beam equation

We recover an unknown space–time-dependent force in an Euler–Bernoulli beam vibration equation by an effective combination of the Lie-group adaptive method (LGAM) and the differential quadrature method (DQM). The layer-stripping technique is used to simplify this identification problem. The DQM is a feasible tool to semi-discretize the Euler–Bernoulli beam equation into a system of ordinary differential equations (ODEs) in time. Then, we can develop a two-point Lie-group equation to recover the unknown force through a few iterations. The success of the present method hinges on a rationale that the local in time ODEs and the global in time algebraic Lie-group equation have to be self-adapted during the iteration processes. The feasibility, accuracy and efficiency of the present method are assessed by comparing the estimated results with some exact solutions.