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In this paper, we treat a numerical scheme for the regular fractional Sturm-Liouville problem containing the Prabhakar fractional derivatives with the mixed boundary conditions. We show that the eigenfunctions corresponding to distinct numerical eigenvalues are orthogonal in the Hilbert spaces. The numerical errors and convergence rates are also investigated. Further, we consider a space-fractional diffusion equation and study the associated fractional Sturm-Liouville problem along with the convergence analysis.

In this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

Background: The Sydney system offers a standard biopsy protocol for detection and follow-up of gastric preneoplastic lesions such as intestinal metaplasia (IM). The highest frequency of cardia-type gastric adenocarcinoma (GA) in Iran has been documented in the north-western part of the country. This study aims to investigate the effect of the addition of mucosal biopsies of gastric cardia to the standard Sydney protocol on the rate of detection of IM in the asymptomatic residents of this high-risk region for proximal gastric cancer. Methods: A retrospective new analysis was performed on the previous data obtained in cross-sectional endoscopic screening in 2000 as well as a biopsy study of 508 asymptomatic volunteer residents in Meshkinshahr district, Ardabil province. The screening study was conducted in a group of residents aged 40 years and older who did not have any previous GI or hemodynamic problems. Results: Intestinal metaplasia at the Sydney protocol sampling sites was detected in 107 samples belonging to 76 of the 508 (14.99%) volunteers. Twenty-one patients had IM at the cardia. Of these, five patients had IM-cardia (IM only at the cardia). Therefore, adding a cardia biopsy to the set of biopsies diagnosed five more IM cases which were not diagnosed on the standard Sydney protocol (P=0.062). Conclusion: The addition of a biopsy from the cardia to the Sydney protocol biopsy set does not seem to improve the frequency of detection of IM in the residents of this high-risk geographic area for proximal gastric carcinoma.