A DIRICHLET-TYPE PROBLEM FOR AN ELLIPTIC EQUATION WITH A SINGULAR COEFFICIENT IN THE FIRST QUADRANT
Keywords:
Dirichlet-type problem (Dirixle tipidagi masala) Elliptic equation (Elliptik tenglama) Singular coefficient (Singulyar koeffitsiyent) Hankel integral transform (Xankel integral almashtirishi) Bessel function (Bessel funksiyasi) ,Energy integrals method (Energiya integrallari usuli)Abstract
Ushbu ishda birinchi chorakda bitta singulyar koeffitsiyentga ega bo'lgan elliptik turdagi tenglama uchun Dirixle tipidagi masala tadqiq etilgan. Qo'yilgan masalaning muntazam yechimlar sinfida yagona yechilishga egaligi isbotlangan. Masala yechimining yagonaligi energiya integrallari usuli bilan, yechimning mavjudligi esa Xankel integral almashtirishi usuli yordamida ko'rsatilgan. Tadqiqot natijasida yechim berilgan chegaraviy funksiyalar orqali aniq integral ko'rinishida topilgan.
In this work, a Dirichlet-type problem for an elliptic equation with one singular coefficient in the first quadrant is investigated. The unique solvability of the posed problem is proven in the class of regular solutions. The uniqueness of the solution to the posed problem is proven by the energy integrals method, and the existence of the solution is proven by the Hankel integral transform method. As a result, the solution to the problem was found in explicit form and represented in the form of a double integral through the given boundary function.
