Gadozoda Mirzomurod, Hafizov Hasan Majidovich

Annotation:

The article considers second-order partial differential equation with variable coefficients and initial and boundary conditions in a bounded domain. The method of separation of variables is used to solve the equation, and the separating solution is represented as a product of two functions. Then, by separating the variables, we obtain a homogeneous Euler-type differential equation for one of them, and a –dimensional boundary value problem for the second function. We solve the resulting boundary value problem again using the method of separation of variables. As a result, we obtain one-dimensional problems for eigenvalues and systems of eigenfunctions. Solving these one-dimensional problems, we obtain eigenvalues and a system of eigenfunctions, which is a complete system in space of continuous on a bounded domain. The solution to the problem is presented in the from of a – dimensional monotonic and absolutely convergent series, and this series with these condition is the only classical solution to the problem.

Keywords: partial differential equation of the second degree, solution in a form of the series, arbitrary constants.