Boundary Value Problems for a Mixed Equation of Parabolic-Hyperbolic Type of the Third Order

• Published in: Lobachevskii journal of mathematics Vol. 44; no. 12; pp. 5155 - 5163
• Main Authors: Apakov, Yu. P., Sopuev, A. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2023; Springer Nature B.V
• Subjects: Algebra; Analysis; Boundary value problems; Conjugation; Eigenvalues; Eigenvectors; Fredholm equations; Geometry; Green's functions; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Partial differential equations; Probability Theory and Stochastic Processes; Volterra integral equations; integral equations; differential equation; Green’s function; conjugation problem; uniqueness; multiple characteristics; existence; boundary conditions; third order
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080223120065

In this article, the existence and uniqueness of solution of the conjugation problem in a rectangular domain for a third-order partial differential equation is proved, when the characteristic equation has 3 multiple roots for , and it has 1 simple and 2 multiple roots for . Using the Green’s functions and the method of integral equations, the solution of the problem is equivalently reduced to solving the boundary value problem for the trace of the desired function at , and then to solving the Fredholm integral equation of the 2nd kind. The one-valued solvability of Fredholm integral equation of the 2nd kind is proved by the method of successive approximations. The solution of the problem for is constructed by the Green’s function method, and for by reducing to the problem of a two-dimensional Volterra integral equation of the 2nd kind.