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

Dang Duc Trong; Nguyen Thanh Long; Pham Ngoc Dinh Alain

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)

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

References:

[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