Dang Duc Trong; Nguyen Thanh Long; Pham Ngoc Dinh Alain Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term. (English) Zbl 1087.35095 J. Math. Anal. Appl. 312, No. 1, 93-104 (2005). 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. Reviewer: Dinh Nho Hao (Hanoi) Cited in 39 Documents MSC: 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 Keywords:error estimates; uniqueness; Fourier transform; ill-posed problem PDFBibTeX XMLCite \textit{Dang Duc Trong} et al., J. Math. Anal. Appl. 312, No. 1, 93--104 (2005; Zbl 1087.35095) Full Text: DOI HAL References: [1] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice Hall: 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: V.H. Winston and Sons Washington, DC · Zbl 0354.65028 [3] Payne, L. E., Improperly Posed Problems in PDE (1971), SIAM: SIAM Philadelphia · Zbl 0302.35003 [4] Colton, D.; Kress, R., Integral Equation Methods in Scattering Theory (1983), Wiley: Wiley New York · Zbl 0522.35001 [5] Linz, P., Analytical and Numerical Methods for Volterra Equations (1985), SIAM: 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.