Forward uncertainty quantification in random differential equation systems with delta‐impulsive terms: Theoretical study and applications

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 7; pp. 7609 - 7629
• Main Authors: Bevia, Vicente J., Cortés, Juan C., Villanueva, Rafael J.
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.05.2025
• Subjects: Dirac‐delta impulse terms; first probability density function; Liouville–Gibbs equation; Partial differential equations; Probability density functions; random differential equations; random variable transformation technique; Random variables; Statistical analysis; Stochastic models; Stochastic processes
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9226

This contribution aims at studying a general class of random differential equations with Dirac‐delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so‐called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.