Application of the projection-grid method to solve a non-stationary problem

The work is dedicated to the construction of an approximate solution of a parabolic differential equation with a Bessel operator. The solution of the problem is sought in the form of a linear combination of piecewise continuous basis functions having a compact carrier. The construction of the solution is carried out in two stages. Initially, approximation on the spatial variable is carried out using the Bubnov-Galerkin projection-grid method. Then, due to the simplicity of the time variable domain, which is the interval [0,T], an approximation on t is conducted using the finite-difference method. For this purpose, an implicit scheme is used. The resulting system of equations has a tridiagonal matrix and is solved by the run method.