Analogue of the Tricomi problem for the mixed-type equation with fractional derivative. Inverse problems

TL;DR

This paper investigates an analogue of the Tricomi problem for mixed-type equations involving fractional derivatives, focusing on the inverse problem of determining the fractional order, which is addressed for the first time in this context.

Contribution

It introduces a novel approach to determine the fractional derivative parameter in a mixed-type equation with a Tricomi-like problem, establishing conditions for uniqueness and existence.

Findings

Established a condition ensuring uniqueness of the fractional derivative parameter.

Proved the existence of solutions under the new condition.

First study of fractional derivative determination in Tricomi-type problems.

Abstract

In this work, an analogue of the Tricomi problem for equations of mixed type with a fractional derivative is investigated. In one part of the domain, the considered equation is a subdiffusion equation with a fractional derivative of order ? 2 (0; 1) in the sense of Riemann-Liouville, and in the other it is a wave equation. Assuming the parameter ? to be unknown, the corresponding inverse problem is studied . It was found an additional condition, that provides not only uniqueness but also existance of the desired parameter. It should be noted that the inverse problem of determining the fractional derivative for the subdiffusion and wave equations has been studied by many mathematicians. But in the case of the Tricomi problem for a mixed-type equation, the questions of determining the fractional time derivative are studied for the first time.