Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.1088/1751-8121/aa763a
· OA: W2605762077
In this article, we consider the reconstruction of $\\rho(t)$ in the\n(time-fractional) diffusion equation\n$(\\partial_t^\\alpha-\\triangle)u(x,t)=\\rho(t)g(x)$ ($0<\\alpha \\le 1$) by the\nobservation at a single point $x_0$. We are mainly concerned with the situation\nof $x_0 \\notin$ supp g, which is practically important but has not been well\ninvestigated in literature. Assuming the finite sign changes of $\\rho$ and an\nextra observation interval, we establish the multiple logarithmic stability for\nthe problem based on a reverse convolution inequality and a lower estimate for\npositive solutions. Meanwhile, we develop a fixed point iteration for the\nnumerical reconstruction and prove its convergence. The performance of the\nproposed method is illustrated by several numerical examples.\n
