On Solvability of a Poincare–Tricomi Type Problem for an Elliptic–Hyperbolic Equation of the Second Kind

• Published in: Lobachevskii journal of mathematics Vol. 42; no. 3; pp. 663 - 675
• Main Authors: Yuldashev, T. K., Islomov, B. I., Abdullaev, A. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.03.2021; Springer Nature B.V
• Subjects: Algebra; Analysis; Boundary value problems; Dirichlet problem; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Partial differential equations; Probability Theory and Stochastic Processes; Regularization; Regularization methods; Singular integral equations; energy integral method; Green function; Poincare–Tricomi type problem; degenerate equation of the second kind; generalized solution
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080221030239

In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.