Problems with Missing Tricomi Condition and Analog of Frankl Condition for One Class of Mixed Type Equations

Assel Makulbay, Mirakhmat Mirsaburov (), Abdumauvlen Berdyshev () and Gulbakhor Mirsaburova
Additional contact information
Assel Makulbay: Department of Mathematics and Mathematical Modelling, Abai Kazakh National Pedagogical University, Almaty 050010, Kazakhstan
Mirakhmat Mirsaburov: Department of Mathematical Analysis, Termez State University, Termez 190111, Uzbekistan
Abdumauvlen Berdyshev: Department of Mathematics and Mathematical Modelling, Abai Kazakh National Pedagogical University, Almaty 050010, Kazakhstan
Gulbakhor Mirsaburova: Department of General Mathematics, Tashkent State Pedagogical University, Tashkent 100185, Uzbekistan

Mathematics, 2025, vol. 13, issue 11, 1-35

Abstract: In this paper, for a mixed elliptic-hyperbolic type equation with various degeneration orders and singular coefficients, theorems of uniqueness and existence of the solution to the problem with a missing Tricomi condition on boundary characteristic and with an analog of Frankl condition on different parts of the cut boundary along the degeneration segment in the mixed domain are proved. On the degeneration line segment, a general conjugation condition is set, and on the boundary of the elliptic domain and degeneration segment, the Bitsadze–Samarskii condition is posed. The considered problem, based on integral representations of the solution to the Dirichlet problem (in elliptic part of the domain) and a modified Cauchy problem (in hyperbolic part of the domain), is reduced to solving a non-standard singular Tricomi integral equation with a non-Fredholm integral operator (featuring an isolated first-order singularity in the kernel) in non-characteristic part of the equation. Non-standard approaches are applied here in constructing the solution algorithm. Through successive applications of the theory of singular integral equations and then the Wiener–Hopf equation theory, the non-standard singular Tricomi integral equation is reduced to a Fredholm integral equation of the second kind, the unique solvability of which follows from the uniqueness theorem for the problem.

Keywords: mixed-type equation with various degeneration orders; missing Tricomi condition; Bitsadze-Samarskii condition; Frankl condition analogue; Wiener–Hopf equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/11/1875/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/11/1875/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:11:p:1875-:d:1671314

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().