Premium
Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense
Author(s) -
Cortés Juan Carlos,
Jornet Marc
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6921
Subject(s) - mathematics , series (stratigraphy) , lebesgue integration , random field , diffusion , random variable , constant (computer programming) , diffusion equation , combinatorics , discrete mathematics , mathematical analysis , statistics , quantum mechanics , physics , paleontology , computer science , biology , programming language , economy , economics , service (business)
In this paper, we deal with the randomized generalized diffusion equation with delay: u t ( t , x ) = a 2 u x x ( t , x ) + b 2 u x x ( t − τ , x ) , t > τ , 0 ≤ x ≤ l ; u ( t , 0 ) = u ( t , l ) = 0 , t ≥ 0 ; u ( t , x ) = φ ( t , x ) , 0 ≤ t ≤ τ , 0 ≤ x ≤ l . Here, τ > 0 and l > 0 are constant. The coefficients a 2 and b 2 are nonnegative random variables, and the initial condition φ ( t , x ) and the solution u ( t , x ) are random fields. The separation of variables method develops a formal series solution. We prove that the series satisfies the delay diffusion problem in the random Lebesgue sense rigorously. By truncating the series, the expectation and the variance of the random‐field solution can be approximated.