The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients

• Main Authors: Akhmadzhon K. Urinov, Kamoliddin T. Karimov
• Format: Article
• Language: English
• Published: Samara State Technical University 2017-12-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Subjects: Dirichlet problem; mixed-type equations; spectral method; uniqueness of solution; existence of solution
• Online Access: http://mi.mathnet.ru/eng/vsgtu1559

We study the Dirichlet problem in a parallelepiped for a three-dimensional equation of mixed type with three singular coefficients. Separation of variables with Fourier series and spectral analysis are used to investigate this problem. Two one-dimensional spectral problems are obtained for the possed problem using the Fourier method. On the basis of the completeness property of the eigenfunction systems of these problems, the uniqueness theorem is proved. The solution of the problem is constructed as the sum of a double Fourier–Bessel series. In justification of the uniform convergence of the series constructed, asymptotic estimates of the Bessel functions of the real and imaginary argument are used. On their basis, estimates are obtained for each member of the series. The estimates obtained made it possible to prove the convergence of the series and its derivatives up to the second order inclusive, and also the existence theorem in the class of regular solutions.