# Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

## Abstract

[EN] In this paper, we deal with the randomized generalized diffusion equation with delay:u(t)(t, x) = a(2)u(xx)(t, x) + b(2)u(xx)(t - tau, x),t > tau,0 = 0;u(t,x)=phi(t,x),0 0andl > 0are constant. The coefficientsa(2)andb(2)are nonnegative random variables, and the initial condition phi(t, x)and the solutionu(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.Secretaria de Estado de Investigacion, Desarrollo e Innovacion, Grant/Award Number: MTM2017-89664-P; Spanish Ministerio de Economia, Industria y Competitividad (MINECO); Agencia Estatal de Investigacion (AEI); Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: MTM2017-89664-PCortés, J.; Jornet, M. (2021). 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