This paper presents searches for flaring sources of neutrinos using the IceCube neutrino telescope. For the first time, a search is performed over the entire parameter space of energy, direction and time looking for neutrino flares of 20 microseconds to a year duration from astrophysical sources among the atmospheric neutrino and muon backgrounds. Searches which integrate over time are less sensitive to flares because they are affected by a larger background of atmospheric neutrinos and muons that can be reduced by the time constraint. Flaring sources considered here, such as active galactic nuclei, soft gamma-ray repeaters and gamma-ray bursts, are promising candidate neutrino emitters. We used mainly data taken between April 5, 2008 and May 20, 2009 by a partially completed configuration of IceCube with 40 strings. For the presented searches an unbinned maximum likelihood method is used with a time-dependent prior to test several different source hypotheses. An untriggered search covers any possible time-dependent emission from sources not correlated to any other observation using other astrophysical messengers such as photons. Moreover, a similar time scan is performed for a predefined catalogue of sources that exhibit intense photon flares. Searches triggered by multi-wavelength information on flares from blazars and soft gamma-ray repeaters are performed using the 40 string data and also the data taken by the previous configuration of 22 strings in operation between May 31, 2007 and April 5, 2008. Flares for which extensive and continuous monitoring is available from Fermi-LAT and SWIFT and flares detected by imaging Cherenkov telescopes with shorter time-scale monitoring are considered. The results from all searches are compatible with a fluctuation of the background.
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