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In this paper, a second finite difference method on a graded grid is proposed for a Volterra integro-differential equation with a weakly singular kernel. The proposed scheme is obtained by using the two-step backward differentiation formula (BDF2) to discretize the first derivative term and the first-order interpolation scheme to approximate the integral term. The analysis of stability is proved and used to prove the convergence of our presented numerical method in the discrete maximum norm. Finally, Numerical experiments are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method. [ABSTRACT FROM AUTHOR]

A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation. [ABSTRACT FROM AUTHOR]

Amidst rising Parkinson's disease (PD) incidence in an aging global population, the need for non-invasive and reliable diagnostic methods is increasingly critical. This review evaluates the strategic role of transcranial sonography (TCS) in the early detection and monitoring of PD. TCS's ability to detect substantia nigra hyperechogenicity offers profound insights into its correlation with essential neuropathological alterations-namely, iron accumulation, neuromelanin depletion, and glial proliferation-fundamental to PD's pathophysiology. Our analysis highlights TCS's advantages, including its non-invasiveness, cost-effectiveness, and ease of use, positioning it as an invaluable tool for early diagnosis and continual disease progression monitoring. Moreover, TCS assists in identifying potential risk and protective factors, facilitating tailored therapeutic strategies to enhance clinical outcomes. This review advocates expanding TCS utilization and further research to maximize its diagnostic and prognostic potential in PD management, contributing to a more nuanced understanding of the disease., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2024 Zhang, Jiang, Zhu, Zhuo and Liu.)

Objective: The chemokine receptor CX3CR1 and its specific ligand fractalkine (CX3CL1, FKN) has been implicated in modulating inflammatory and fibroproliferative diseases. The current study was performed to investigate the correlation of serum fractalkine levels with disease severity of liver fibrosis/cirrhosis (LC)., Methods: 162 LC patients and 140 healthy controls well enrolled in our study. Serum fractalkine levels were detected using commercial ELISA kit. Liver biopsy specimens were obtained using 16 G disposable needle in LC patients. The Child-Pugh grade was recorded to assess liver function. ROC curve analysis was performed to assess the potential diagnostic power of serum fractalkine with regard to the disease severity of Child-Pugh grade system. Pathological assessment of cirrhotic severity was performed by Laennec staging system. The L3 skeletal muscle index (L3SMI) was applicated to assess the nutrition status., Results: Serum fractalkine levels were significantly higher in LC patients compared with healthy controls. The case group included 50 Child-Pugh A patients, 59 Child-Pugh B patients, and 53 Child-Pugh C patients. Cirrhosis patients with Child-Pugh C had drastically higher serum fractalkine levels compared with those with Child-Pugh B and A. Child-Pugh B patients showed significantly higher serum PACAP concentrations compared with those with Child-Pugh A. ROC curve analysis demonstrated that serum fractalkine may act as a potential indicator for disease progression of LC determined by Child-Pugh classification. Besides, serum fractalkine levels were positively related to ALT and AST concentrations and negatively related to L3SMI., Conclusion: Serum fractalkine levels were positively associated with disease severity of LC., (© 2021 by the Association of Clinical Scientists, Inc.)