Approximate solution of the main boundary value problem for a polygarmonic equation in the ring-shaped domain

Background. This work is devoted to the actual problem of construction and development of effective numerical methods to solve a polyharmonic equation. The aim of the paper is to obtain an approximate solution of the basic boundary-value problem for a polyharmonic equation in a doubly-connected domain, bounded from the inside by contour $\partial D_1$ and from the outside by contour $\partial D_2$ (ring-shaped domain). Materials and methods. The problem is solved by using the conformal mapping of the considered domain to a circular ring. The desired n -harmonic function is represented by n analytic functions of a complex variable, each of which is sought in the circular ring in the form of a Laurent series. To calculate the coefficients of the series a numerical collocation method is applied. Results. An approximate numerical-analytic solution of the main boundary-value problem for a polyharmonic equation in the ring-shaped domain is obtained. The test examples are considered and confirm the good accuracy of the solution. Conclusions. From the test examples it can be seen that the proposed method for solving the main boundary-value problem for a polyharmonic equation in the ring-shaped domain is quite effective.