A modified Cauchy problem for an inhomogeneous equation of degenerate hyperbolic type of the second kind

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Abstract

In this study, a modified Cauchy problem was examined for an inhomogeneous equation of degenerate hyperbolic type of the second kind in a characteristic triangle. It is known that degenerate hyperbolic equations have a singularity, meaning that the well-posedness of the Cauchy problem with initial data on the line of parabolic degeneracy does not always hold for them. Therefore, in such cases, it is necessary to consider the problem with initial conditions in a modified form.
In present paper, modified Cauchy problems with initial conditions were formulated on the line of parabolic degeneracy for an inhomogeneous equation of degenerate hyperbolic type of the second kind. The considered problem is reduced to a modified Cauchy problem for a homogeneous equation and to a Cauchy problem for an inhomogeneous equation with zero initial conditions. The solutions of the modified Cauchy problem for a homogeneous equation are derived from the general solution of the considered equation. The explicit solutions of the modified Cauchy problem with homogeneous conditions for the inhomogeneous equation are found using the Riemann
method.
It is proven that the discovered solutions indeed satisfy the equation and the initial conditions.

About the authors

Akhmadjon K. Urinov

Fergana State University; V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan

Email: urinovak@mail.com
ORCID iD: 0000-0002-9586-1799
Scopus Author ID: 19639412400
https://www.mathnet.ru/person30024

Dr. Phys. & Math. Sci., Professor; Head of Department; Dept. of Mathematical Analysis and Differential Equations; Leading Researcher

Uzbekistan, 150100, Ferghana, Murabbiylar st., 19; 100174, Tashkent, University st., 46

Akmaljon B. Okboev

V.I. Romanovskiy Institute of Mathematics
of the Academy of Sciences of the Republic of Uzbekistan

Author for correspondence.
Email: akmaljon12012@gmail.com
ORCID iD: 0000-0002-5544-3111
Scopus Author ID: 57216342298
https://www.mathnet.ru/person117934

PhD (Phys. & Math. Sci.); Senior Researcher

Uzbekistan, 100174, Tashkent, University st., 46

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