Closed-form higher-order numerical differentiators for differentiating noisy signals

In the paper, nth-order differential of a noisy signal is recast as an nth-order ordinary differential equation with an unknown right-hand side, which is an inverse problem to recover the forcing term. We derive weak-form methods to solve the inverse problem, with sinusoidal functions as test functions. By exploring the orthogonality of sinusoidal functions, the expansion coefficients in the trial functions of weak-form numerical differentiators can be determined analytically. Several examples verify the efficiency, accuracy and robustness of the weak-form numerical differentiators for computing the higher-order differentials of noisy data. Moreover, the applications of the weak-form numerical differentiators are also demonstrated, to recover the external forces of nonlinear dynamical systems with single or multiple degrees of freedoms, which are evaluated under the pollution of large noise on the measured data of displacements.