Geodesic
A nonhomogeneous backward heat problem: regularization and error estimates
Electronic Journal of Differential Equations, Tome 2008 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We consider the problem of finding the initial temperature, from the final temperature, in the nonhomogeneous heat equation $$\displaylines{ u_t-u_{xx}= f(x,t),\quad (x,t)\in (0,\pi)\times (0,T),\cr u(0,t)= u(\pi,t)= 0, \quad (x,t) \in (0,\pi)\times(0,T). }$$ This problem is known as the backward heat problem and is severely ill-posed. Our goal is to present a simple and convenient regularization method, and sharp error estimates for its approximate solutions. We illustrate our results with a numerical example.
Classification : 35K05, 35K99, 47J06, 47H10
Keywords: backward heat problem, ill-posed problem, nonhomogeneous heat equation, contraction principle 
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     author = {Dang Duc Trong and Nguyen Huy Tuan},
     title = {A nonhomogeneous backward heat problem: regularization and error estimates},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2008},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a73/}
}
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Dang Duc Trong; Nguyen Huy Tuan. A nonhomogeneous backward heat problem: regularization and error estimates. Electronic Journal of Differential Equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a73/