## 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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