APPROXIMATE SOLUTION OF THE MAIN BOUNDARY VALUE PROBLEM FOR A POLYGARMONIC EQUATION IN THE RING-SHAPED DOMAIN 

Kazakova Anastasiya Olegovna, Candidate of physical and mathematical sciences, associate professor, sub-department of actuarial and financialmathematics, Chuvash State University named after I. N. Ulianov (15 Moskovskiy avenue, Cheboksary, Russia), kazakova_anastasia@bk.ru

517.95

10.21685/2072-3040-2017-3-2

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 D1 and from the outside by contour D2 (ring-shaped domain).
Materials and methods. The problem is solved by using the conformal mappingof 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 inthe circular ring in the form of a Laurent series. To calculate the coefficients of theseries a numerical collocation method is applied.
Results. An approximate numerical-analytic solution of the main boundary-valueproblem for a polyharmonic equation in the ring-shaped domain is obtained. Thetest examples are considered and confirm the good accuracy of the solution.
Conclusions. From the test examples it can be seen that the proposed method forsolving the main boundary-value problem for a polyharmonic equation in the ringshapeddomain is quite effective.

Laplace operator, polyharmonic equation, main boundary value problem, doubly-connected ring-shaped domain, conformal mapping, Laurent series, collocation method, system of linear algebraic equations

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