We consider the fourth-order differential theory of gravitation to treat the problem of singularity avoidance: studying the short-distance behaviour in the case of black-holes and the big-bang we are going to see a way to attack the issue from a general perspective.
The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.
Investigation of exceptional points mostly focuses on the second order case and employs the gain-involved parity-time (PT) symmetric systems. Here, we propose an approach to implementing fourth order exceptional points (FOEPs) using directly coupled optical resonators with rotation. On resonance, the system manifests FOEP through tuning the spinning velocity to targeted values. Eigenfrequency bifurcation and enhanced sensitivity for nanoparticle have been presented. Also, near FOEP, nonreciprocal light propagation exhibits great boost and dramatic change, which may be applied to high-efficiency isolators and circulators.