Abstract
We are concerned with the reconstruction of the thermal conductivity coefficient in a one-dimensional heat equation from observations of solutions at a single point in space and time. To this end, we combine spectral estimation together with asymptotics of the solution to extract the Fourier coefficients of the first eigenfunction. In the case of Neumann boundary conditions, we need to reconstruct the second eigenfunction. Once we have obtained an eigenfunction, the sought conductivity coefficient is obtained as a solution of a simple differential equation. This direct method produces a fast algorithm which is an alternative to the Gelfand-Levitan inverse spectral theory, minimization procedures, or fixed point methods.
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Boumenir, A. The reconstruction of the space-dependent thermal conductivity coefficient. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01724-5
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DOI: https://doi.org/10.1007/s11075-023-01724-5