Interpolation Method of Shalva Mikeladze with Following Applications

Abstract

The paper deals with the numerical method of Shalva Mikeladze, the accuracy of which depends on the number of interpolation points. The method called the method without saturation is devoted to the numerical solution of ordinary differential equations. It is constructed on the basis of an interpolation formula to solve numerically linear and nonlinear ordinary differential equations of any order and systems of such equations. Using its different versions, it is possible to solve boundary value, eigenvalue, and Cauchy problems (Mikeladze, Soobsh AN GSSR 45(2):284–296, 1967 and Mikeladze, Soobsh AN GSSR 47(2):263–268, 1967). This method in combination with the method of lines can also be applied to solve boundary value problems for partial differential equations of elliptic type (Makarov, Karalashvili, Soobsh AN GSSR 131(1):33–36, 1988). As a model, the Dirichlet problem for a Poisson equation in the symmetric rectangle is considered. This application created a semi-discrete difference scheme with matrices of central symmetry having certain properties.