Time-varying linear equation systems have been solved by the traditional zeroing neural dynamics approach in recent years. However, this method has to satisfy the stability constraint, which is a very rigorous condition. For this reason, traditional Lagrange-type finite difference formulas fail to lead to effective solutions, and we have to utilize more instants and reduce accuracy so that this condition is satisfied. In this work, we develop a new method of solving a time-varying linear equation system, which is based on theoretical solution decomposition. As a result, the proposed solutions are stability-constraint-free. We do not have to meticulously search effective time-discretization formulas because traditional Lagrange-type formulas are sufficient and especially effective. In addition, the proposed solutions have other advantages. For example, they do not need convergence procedures; they do not have storage requirements for past calculative results; and they are still effective when the sampling gap value is relatively large. Detailed comparisons are presented in this paper. Comparative numerical experiments are also shown to substantiate the effectiveness and advantages of the proposed solutions.

Resource allocation problem in a wireless sensor network can be formulated as time-varying optimization, which can be further converted as time-varying linear equation system (TVLES). Hybrid multilayered time-varying linear equation system (HMTVLES) involving hybrid multilayers and time-variation characteristic is a complicated and challenging problem. Recently it has been solved by zeroing neural dynamics (ZND) method under ideal conditions, i.e., without noises. However, noises are ubiquitous, immanent and unavoidable in real-time systems. In this work, we propose a noise-tolerant zeroing neural dynamics (NTZND) model for solving HMTVLES. It can deal with different kinds of noises such as constant noise, linear-increasing noise, and random noise. Theoretical analyses guarantee the precision of NTZND model in the presence of different kinds of noises. In addition, a general NTZND model is proposed based on general activation function. Besides, classical ZND method and gradient neural dynamics (GND) method are also investigated and compared. Numerical experimental results are presented to verify the theoretical results of proposed models.