In this talk, we will report a temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. At first, a special point is found for the interpolation approximation of the linear combination of multi-term fractional derivatives. The derived numerical differentiation formula can achieve over second order accuracy. The formula is then used to numerically solve the time multi-term and distributed-order fractional sub-diffusion equations. Several difference schemes are presented. The unconditional stability and convergence of the difference schemes are discussed. Some numerical examples are provided to show the efficiency of the proposed difference schemes.

The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-order Fractional Sub-diffusion Equations

Diabetic wound is one of the complications of diabetes and is not easy to heal. It often evolves into chronic ulcers, and severe patients will face amputation. Compared with normal wounds, diabetic wounds have an increased proportion of pro-inflammatory cytokines that are detrimental to the normal healing response. The burden of this disease on patients and healthcare providers is overwhelming, and practical solutions for managing and treating diabetic wounds are urgently needed. Pyroptosis, an inflammatory type of programmed cell death, is usually triggered by the inflammasome. The pyroptosis-driven cell death process is primarily mediated by the traditional signaling pathway caused by caspase -1 and the non-classical signaling pathways induced by caspase -4/5/11. Growing evidence that pyroptosis promotes diabetic complications, including diabetic wounds. In addition, inflammation is thought to be detrimental to wound healing. It is worth noting that the activation of the NLRP3 inflammasome plays a crucial role in the recovery of diabetic wounds. This review has described the mechanisms of pyroptosis-related signaling pathways and their impact on diabetic wounds. It has discussed new theories and approaches to promote diabetic wound healing, as well as some potential compounds targeting pyroptosis and inflammasome signaling pathways that could be new approaches to treating diabetic wounds.

https://www.frontiersin.org/articles/10.3389/fendo.2022.950798/pdf

Pyroptosis and inflammasomes in diabetic wound healing

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in $$L^2$$ -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm. 

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems

Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space

In this paper, we shall present the alternating direction implicit (ADI) Galerkin finite element method (FEM) for solving the distributed-order time-fractional mobile–immobile equation in two dimensions. In the time direction, the backward Euler method is used to deal with the temporal first-order derivative, and the weighted and shifted Grunwald formula is employed to discretize the distributed-order time-fractional derivative. Galerkin FEM is used for discretization of the spatial direction, and then an ADI Galerkin finite element scheme is constructed. The stability and L 2 -norm convergence are proved. Several numerical experiments are provided to verify our theoretical analysis.

An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile–immobile equation in two dimensions

/pdf/second-order-flows-for-computing-the-ground-states-of-3en6fphm.pdf

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

Porcine reproductive and respiratory syndrome virus (PRRSV) is one of the pathogens responsible for major economic losses in the global swine industry. Characterizing the detailed process by which PRRSV induces cell death pathways will help us better understand viral pathogenesis and provide implications for therapeutic intervention against PRRSV. ABSTRACT For more than 3 decades, mounting evidence has associated porcine reproductive and respiratory syndrome virus (PRRSV) infection with late-term abortions and stillbirths in sows and respiratory disease in piglets, causing enormous economic losses to the global swine industry. However, to date, the underlying mechanisms of PRRSV-triggered cell death have not been well clarified, especially in the pulmonary inflammatory injury characterized by the massive release of pro-inflammatory factors. Here, we demonstrated that PRRSV infection triggered gasdermin D-mediated host pyroptosis in vitro and in vivo. Mechanistically, PRRSV infection triggered disassembly of the trans-Golgi network (TGN); the dispersed TGN then acted as a scaffold for NLRP3 activation through phosphatidylinositol-4-phosphate. In addition, PRRSV replication-transcription complex (RTC) formation stimulated TGN dispersion and pyroptotic cell death. Furthermore, our results indicated that TMEM41B, an endoplasmic reticulum (ER)-resident host protein, functioned as a crucial host factor in the formation of PRRSV RTC, which is surrounded by the intermediate filament network. Collectively, these findings uncover new insights into clinical features as previously unrecognized mechanisms for PRRSV-induced pathological effects, which may be conducive to providing treatment options for PRRSV-associated diseases and may be conserved during infection by other highly pathogenic viruses. IMPORTANCE Porcine reproductive and respiratory syndrome virus (PRRSV) is one of the pathogens responsible for major economic losses in the global swine industry. Characterizing the detailed process by which PRRSV induces cell death pathways will help us better understand viral pathogenesis and provide implications for therapeutic intervention against PRRSV. Here, we showed that PRRSV infection induces GSDMD-driven host pyroptosis and IL-1β secretion through NOD-, LRR- and pyrin domain-containing protein 3 (NLRP3) inflammasome activation in vitro and in vivo. Furthermore, the molecular mechanisms of PRRSV-induced NLRP3 inflammasome activation and pyroptosis are elucidated here. The dispersed trans-Golgi network (TGN) induced by PRRSV serves as a scaffold for NLRP3 aggregation into multiple puncta via phosphatidylinositol 4-phosphate (PtdIns4P). Moreover, the formation of PRRSV replication-transcription complex is essential for TGN dispersion and host pyroptosis. This research advances our understanding of the PRRSV-mediated inflammatory response and cell death pathways, paving the way for the development of effective treatments for PRRSV diseases.

PRRSV Infection Induces Gasdermin D-Driven Pyroptosis of Porcine Alveolar Macrophages through NLRP3 Inflammasome Activation

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An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile–immobile equation in two dimensions

Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space

Second-order flows for computing the ground states of rotating Bose-Einstein condensates

PRRSV Infection Induces Gasdermin D-Driven Pyroptosis of Porcine Alveolar Macrophages through NLRP3 Inflammasome Activation