利用弱型式之數值微分運算子重建非線性系統之外力

Abstract

二階微分之噪音訊號被寫為二階常微分方程式，當作一種特殊實例未知外力在二階線性系統之回復，轉換成線性拋物線型的偏微分方程式。之後運用格林第二恆等式推導出以伴隨崔維茲頻譜函數表示之邊界積分方程。我們發現一種弱型式的方法可回復外力，之後，從給定之噪音中發展弱型式二階運算子(WFSOD)計算訊號的二階導數，只有訊號本身是具體指定，不需要此噪音訊號之一階導數，最後，在大時間區間和大噪音之下使用弱型式方法回復非線性系統之外力。

The second-order differential of a noisy signal is written as a second-order ordinary differential equation, being a special case of the recovery of unknown external force in a second-order linear system, which is transformed into a linear parabolic type partial differential equation. Then the Green second identity is employed to derive a boundary integral equation in terms of the adjoint Trefftz spectral functions. We find a weak-form method to recover the external force and then a weak-form second-order differentiator(WFSOD) is developed to compute the second-order differential from a given noisy signal, of which only the signal itself is specified, without needing of its first-order differential. Finally, the weak-form method is used to recover the external forces of nonlinear systems within a large time interval and under a large noise.