The accurate prediction of remaining useful life is a significant issue for ensuring the reliable operation of the system. An adaptive kernel density estimation model for real-time remaining useful life is established considering the dynamic transition of degraded states. Firstly, a time series density peak clustering algorithm suitable for real-time manifold data clustering is proposed, which can efficiently, quickly, and accurately cluster data in different degradation states. Then, different degradation state patterns according to clustering results can be divided. Moreover, the smoothing parameters can be adaptively updated according to the sample density under different degradation modes and an adaptive kernel density remaining useful life estimation model is established. The test of the gearbox verifies the necessity and accuracy of the proposed model by comparison with the remaining useful life predictions of kernel density estimation without considering degraded state transitions.

The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution.

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