## Abstract

In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time t = T. This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

Original language | British English |
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Pages (from-to) | 111-134 |

Number of pages | 24 |

Journal | Evolution Equations and Control Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2017 |

## Keywords

- Backward problem
- Fractional evolution equation
- Regularization