Resumen de Solving 2D boundary-value problems using discrete partial differential operators

Marcin Jaraczewski, Tadeusz Sobczyk

• Purpose – Discrete differential operators of periodic base functions have been examined to solve boundaryvalue problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.

  Design/methodology/approach – This paper presents a promising approach for solving twodimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.

  Findings – An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.

  Research limitations/implications – The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.

  Practical implications – This paper presents a methodology for calculations of the self- and mutualleakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.

  Originality/value – The presented methodology allows us to obtain the magnetic vector potential in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.