## Abstract

A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator ut − ∇· (∂ _{t} ^{1} − ^{α} κα∇u− F∂ _{t} ^{1} − ^{α} u), where 0 < α < 1. The forcing function F = F(t, x), which is more difficult to analyse than the case F = F(x) investigated previously by other authors. The spatial domain Ω ⊂ R ^{d} , where d ≥ 1, has a smooth boundary. Existence, uniqueness and regularity of a mild solution u is proved under the hypothesis that the initial data u _{0} lies in L ^{2} (Ω). For 1/2 < α < 1 and u0 ∈ H ^{2} (Ω) ∩ H _{0} ^{1} (Ω), it is shown that u becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived—these are known to be needed in numerical analyses of this problem.

Original language | English |
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Pages (from-to) | 2765-2787 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Analysis |

Volume | 18 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Sep 2019 |

## Keywords

- Fokker–Planck equation
- Regularity of solution
- Riemann–Liouville derivative
- Time-fractional