Noninvasive positive pressure ventilation (NIPPV) plays an important role in the management of respiratory failure. However, since the emergence of the COVID-19 pandemic, utilization of traditional face mask NIPPV has been curtailed in part due to risk of aerosolization of respiratory particles and subsequent health care worker exposure. A randomized clinical trial in 2016 reported that an alternative interface, helmet NIPPV, may be more effective than traditional NIPPV at preventing intubation and improving mortality. The helmet NIPPV interface provides positive airway pressure, while also theoretically minimizing aerosolization, making it a feasible modality in management of respiratory failure in COVID-19 patients. This report describes a single-center experience of a series of three COVID-19 patients with hypoxemic respiratory failure managed with helmet NIPPV. One patient was able to avoid intubation while a second patient was successfully extubated to NIPPV. Ultimately, the third patient was unable to avoid intubation with helmet NIPPV, although the application of the device was late in the progression of the disease. NIPPV is an important modality in the management of respiratory failure and has been shown to reduce the need for immediate endotracheal intubation in select populations. For patients unable to tolerate facemask NIPPV, the helmet provides an alternate interface. In COVID-19 patients, the helmet interface may reduce the risk of virus exposure to health care workers from aerosolization. Based on this experience, we recommend that helmet NIPPV can be considered as a feasible option for the management of patients with COVID-19, whether the goal is to prevent immediate intubation or avoid post-extubation respiratory failure. Randomized studies are needed to definitively validate the use of helmet NIPPV in this population. Helmet NIPPV is a feasible therapy to manage COVID-19 patients.

The introduction of new linear block method for the direct simulation of fourth order IVPs has been developed in this article. The reason for adopting direct simulation of fourth order initial value problems is to address some setbacks in reduction method. When developing the method, we adopted the linear block approach through a one step method. We have validated the accuracy of the method on some fourth order initial value problems without reduction process, and the results are better than the conventional method. The numerical experiments were given and the results obtained were found to be better in accuracy than the existing methods in literature.

Real-world problems, particularly in the sciences and engineering, are often analyzed using differential equations to understand physical phenomena. Many situations involve rates of change of independent variables, represented by derivatives, which lead to differential equations. Solving higher-order ordinary differential equations typically involves reducing them to systems of first-order equations, but this approach has challenges. To overcome these and enhance numerical methods, a novel one-step block method with eight partitions was developed for the direct solution of higher-order initial value problems. This method will target issues in physics, biology, chemistry and economics. The new method was formulated using the linear block approach and numerical analysis was ensure essential and sufficient conditions. The new method addresses second-order problems like simple harmonic motion, third-order issues such as oscillatory differential equations, and fourth-order problems like thin film dynamics. The new method demonstrates faster convergence and improved accuracy compared to existing solutions for second, third, and fourth-order oscillatory differential equations.

This study demonstrates the derivation of a two-step block scheme simulation through a linear block approach. The scheme's fundamental properties were thoroughly analyzed and found to fulfill all necessary conditions. The research focused on examining specific classes of oscillatory differential equations and comparing them to established methods. The findings indicate that the newly proposed methods exhibit superior accuracy and faster convergence compared to the existing methods investigated in this research. Consequently, the results highlight the improved precision and quicker convergence achieved with the new method. All computations were executed using Maple 18 software

The numerical application of higher order linear block method for the direct solution of fourth order initial value problems was proposed using the linear block algorithm, where the methods applied in block form. The method is zero-stabile, consistent and convergent when analyzing the properties of the method. The mathematical example solved using the method is effective, suitable, and acceptable for solving fourth order initial value problems. The method is also compared with existing work when solving similar systems of differential equation and obviously, the method performs better than those in literature and textual shown.

In this research, we have examined the general block approach for solving higher-order oscillatory differential equations using the linear block approach (LBA). The basic properties of the new method, such as order, error constant, zero-stability, consistency, convergence, linear stability, and region of absolute stability, were also analyzed and satisfied. Some distinct fourth-order oscillatory problems were directly applied to the new method in order to overcome the setbacks of the reduction method. The results obtained were compared with those in the literature, and the new method takes away the burden of solving fourth-order oscillatory differential equations. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantages of the new method is that it does not require much computational burden and is also self-starting. Convergence, Consistency, Direct simulation, Linear stability, Oscillatory differential equations