On a non-local problem for a fractional differential equation of the Boussinesq type

Authors

  • R.R. Ashurov Institute of Mathematics of Uzbekistan, Tashkent, Uzbekistan; University of Tashkent for Applied Sciences, Tashkent, Uzbekistan https://orcid.org/0000-0001-5130-466X
  • Yu.E. Fayziev National University of Uzbekistan, Tashkent, Uzbekistan; University of Exact and Social Sciences, Tashkent, Uzbekistan https://orcid.org/0000-0002-8361-2525
  • M.U. Khudoykulova National University of Uzbekistan, Tashkent, Uzbekistan

DOI:

https://doi.org/10.31489/2024m3/34-45

Keywords:

fractional equation, Caputo derivative, forward and inverse problems, Fourier method

Abstract

In recent years, the fractional partial differential equation of the Boussinesq type has attracted much attention from researchers due to its practical importance. In this paper, we study a non-local problem for the Boussinesq type equation Dtαu(t)+A Dtαu(t)+ν2Au(t) =0, 0<t<T, 1<α<3∕2, where Dtα is the Caputo fractional derivative, and A is an abstract operator. In the classical case, i.e., when α = 2, this problem has been studied previously, and an interesting effect has been discovered: the existence and uniqueness of a solution depend significantly on the length of the time interval and the parameter ν. In this note, we show that in the case of a fractional equation, there is no such effect: a solution of the problem exists and is unique for any T and ν.

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Published

2024-09-30

Issue

2024 (115) 3

Section

MATHEMATICS