Short Communication

On the uniqueness of the solution of the Cauchy problem for the equation of fractional diffusion with Bessel operator

F. G. Khushtova

Institute of Applied Mathematics and Automation, Nal'chik, 360000, Russian Federation

Abstract: In this paper, we consider fractional diffusion equation involving the Bessel operator acting with respect to a spatial variable and the Riemann-Liouville fractional differentiation operator acting with respect to a time variable. When the order of the fractional derivative is unity, and the singularity of the Bessel operator is absent, this equation coincides with the classical heat equation. Earlier, a solution of the Cauchy problem has been considered for the considered equation and a uniqueness theorem has been proved for a class of functions satisfying the analog of the Tikhonov condition.
In this paper, we have constructed an example to show that the exponent (power) at the condition of the uniqueness of the solution to the Cauchy problem cannot be raised under. Its increase leads to a non-uniqueness of the solution. Using the well-known properties of the Wright function, we have obtained estimates for constructed function, which satisfies the homogeneous equation and the zero Cauchy condition.

Keywords: fractional diffusion equation, fractional differentiation operator, Bessel operator, Cauchy problem, solution uniqueness, Tikhonov condition, Wright function.

UDC: 517.955, 517.968.7

MSC: 26A33, 35K15, 35R11

Received: August 28, 2018
Revised: October 25, 2018
Accepted: November 12, 2018
First online: November 28, 2018

DOI: 10.14498/vsgtu1639

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