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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.

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
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References:

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