Abstract
The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, x, p(t)) in Q T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) on ∂Ω×(0, T] and either ∫G(t) Φ(x,t)u(x,t)dx = E(t), 0 ⩽ t ⩽ T or u(x0, t)=E(t), 0≤t≤T, where Ω∋R n is a bounded domain with smooth boundary ∂Ω, x 0∈Ω, L is a linear elliptic operator, G(t)∋Ω, and F, ø, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.
Sommario
Gli Autori considerano il problema di trovare u=u(x, t) e p=p(t) che soddisfano u = Lu + p(t) + F(x, t, u, x, p(t)) in Q T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) su ∂Ω×(0, T] e ∫G(t) Φ(x,t)u(x,t)dx = E(t), 0 ⩽ t ⩽ T oppure u(x0, t)=E(t), 0≤t≤T, dove Ω∋R n è un dominic limitato con contorno liscio ∂Ω, x 0∈Ω, L è un operatore lineare allittico, G(t)∋Ω, e F, ø, g, e E sono funzioni note. Per ognuno dei due problemi sopra citati si dimostra l'esistenza, l'unicità e la dipendenza continua dai dati. Sono inoltre presentate alcune considerazioni accompagnate da esempi sulla soluzione numerica di questi due problemi inversi.
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Cannon, J.R., Lin, Y. & Wang, S. Determination of source parameter in parabolic equations. Meccanica 27, 85–94 (1992). https://doi.org/10.1007/BF00420586
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DOI: https://doi.org/10.1007/BF00420586