# Determine the spacial term of a two-dimensional heat source

Abstract : We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t -\Delta u =\varphi(t)f (x,y)$, where $(x,y)\in \Omega=(0,1)\times (0,1)$ and the function $\varphi$ is given. The problem is ill-posed. Under a slight condition on $\varphi$, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term $f$ from non-smooth data. The error estimate and numerical experiments are given.
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Cited literature [15 references]

https://hal.archives-ouvertes.fr/hal-00294612
Contributor : Alain Pham Ngoc Dinh <>
Submitted on : Friday, July 11, 2008 - 11:27:25 AM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Document(s) archivé(s) le : Saturday, November 26, 2016 - 12:22:00 AM

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heatsource2d.pdf
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### Identifiers

• HAL Id : hal-00294612, version 3
• ARXIV : 0807.1806

### Citation

Dang Duc Trong, Alain Pham Ngoc Dinh, Phan Thanh Nam. Determine the spacial term of a two-dimensional heat source. Applicable Analysis, Taylor & Francis, 2009, 88 (3), pp.457-474. ⟨hal-00294612v3⟩

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