Two-Step Relaxation Newton Method for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory

We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form

Some New Results on the Lotka-Volterra System with Variable Delay

This paper discusses the stochastic Lotka-Volterra system with time-varying delay. The nonexplosion, the boundedness, and the polynomial pathwise growth of the solution are determined once and for all by the same criterion. Moreover, this criterion is constructed by the parameters of the system itself, without any uncertain one. A two-dimensional stochastic delay Lotka-Volterra model is taken as an example to illustrate the effectiveness of our result

The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results

Solar Radio Bursts with Spectral Fine Structures in Preflares

A good observation of preflare activities is important for us to understand the origin and triggering mechanism of solar flares, and to predict the occurrence of solar flares. This work presents the characteristics of microwave spectral fine structures as preflare activities of four solar flares observed by Ond\v{r}ejov radio spectrograph in the frequency range of 0.8--2.0 GHz. We found that these microwave bursts which occurred 1--4 minutes before the onset of flares have spectral fine structures with relatively weak intensities and very short timescales. They include microwave quasi-periodic pulsations (QPP) with very short period of 0.1-0.3 s and dot bursts with millisecond timescales and narrow frequency bandwidths. Accompanying these microwave bursts, there are filament motions, plasma ejection or loop brightening on the EUV imaging observations and non-thermal hard X-ray emission enhancements observed by RHESSI. These facts may reveal certain independent non-thermal energy releasing processes and particle acceleration before the onset of solar flares. They may be conducive to understand the nature of solar flares and predict their occurrence

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings