Semi-Plenária 02: Juan Carlos Cortés, Universidad Politecnica de Valencia, Espanha

Título: A Journey to PDFs Methods for Differential Equations with Uncertainties: Bridging Theory and Real-World Applications

Resumo: This talk invites non-specialists on a journey into the fascinating world of differential equations with uncertainty (DEwUs). While deterministic differential equations are powerful tools across science and engineering, they often overlook the unavoidable randomness present in real-world problems -such as noise in measurements, uncertain parameters, or imprecise initial data-. How should we rigorously formulate and solve such equations under uncertainty? We will explore various approaches to modeling randomness in differential equations and highlight why solving a DEwU requires more than just finding a solution. Since solutions are now stochastic processes, understanding their statistical behavior -like mean, variance, or even full probability distributions- is essential. In particular, we will focus on methods for computing the one-point probability density function (1-PDF), a key quantity for capturing the full uncertainty of a solution at a fixed point in time or space. The presentation will highlight how the suitability of each approach depends on the specific nature of the differential equation being studied. Real-world examples will illustrate how computing the 1-PDF can offer insights beyond classical deterministic analysis.