{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:25:59Z","timestamp":1760239559199,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T00:00:00Z","timestamp":1605830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","award":["A1-S-45928"],"award-info":[{"award-number":["A1-S-45928"]}],"id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection\u2013diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.<\/jats:p>","DOI":"10.3390\/sym12111907","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T09:46:18Z","timestamp":1605865578000},"page":"1907","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7580-7533","authenticated-orcid":false,"given":"Jorge E.","family":"Mac\u00edas-D\u00edaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia"},{"name":"Departamento de Matem\u00e1ticas y F\u00edsica, Universidad Aut\u00f3noma de Aguascalientes, Aguascalientes 20131, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,20]]},"reference":[{"key":"ref_1","first-page":"315","article-title":"Almost sure behaviour of sums of independent random variables and martingales","volume":"Volume 2","author":"Strassen","year":"1967","journal-title":"Proceedings of the Fifth Berkeley Symposium"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/BF00539832","article-title":"The tangent approximation to one-sided Brownian exit densities","volume":"61","author":"Ferebee","year":"1982","journal-title":"Z. 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