Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term. (English) Zbl 1087.35095

The problem of finding the pair of functions \(\{u(x,t),f(x)\}\) from the system \(-\partial u/\partial t + u_{xx} = \varphi(t)f(x),\) \((x,t) \in (0,1)\times(0,1),\quad u(1,t) =0, u_x(0,t)=u_x(1,t) =0, u(x,0) =0, u(x,1) = g(x)\) with \(\varphi\) and \(g\) given, is considered. It is proved that for \(\varphi \not\equiv 0\) the solution is unique. Under various assumptions on \(\varphi\) and \(g\) two regularizations based on the Fourier transform associated with a Lebesgue measure for this ill-posed problem are suggested.


35R30 Inverse problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
Full Text: DOI HAL


[1] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice Hall Englewood Cliffs, NJ · Zbl 0144.34903
[2] Tikhonov, A.N.; Arsenin, V.Y., Solutions of ill-posed problems, (1977), V.H. Winston and Sons Washington, DC · Zbl 0354.65028
[3] Payne, L.E., Improperly posed problems in PDE, (1971), SIAM Philadelphia · Zbl 0302.35003
[4] Colton, D.; Kress, R., Integral equation methods in scattering theory, (1983), Wiley New York · Zbl 0522.35001
[5] Linz, P., Analytical and numerical methods for Volterra equations, (1985), SIAM Philadelphia · Zbl 0566.65094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.