Application of discrete differential operators of periodic functions to solve 1D boundary-value problems

Abstract

<jats:sec> <jats:title content-type="abstract-subheading">Purpose</jats:title> <jats:p>Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Design/methodology/approach</jats:title> <jats:p>The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Findings</jats:title> <jats:p>An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Research limitations/implications</jats:title> <jats:p>The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Practical implications</jats:title> <jats:p>Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Originality/value</jats:title> <jats:p>New finite-difference expressions for the first- and second-order derivatives have been applied.</jats:p> </jats:sec>

On Simplified Calculations of Leakage Inductances of Power Transformers

“…Those straight lines have to have strictly the same but opposite derivatives, so the magnetic potential curve has its maximum exactly in the middle of the winding, i.e., the magnetic field takes the zero value at that point. The method of images, presented in [25,26], can be used to solve the case of only one winding in the air zone. The interval of analysis is extended to −d < Figure 6.…”

“…It allows us to find a modified discrete difference operator (DDO) of the second order D (2) , as presented in [25,26], binding the values of the second derivatives and the function itself at the chosen point set {α n } a = D (2) • a (11) where:…”

“…This paper presents an approach for determining inductances L σ n,m for symmetrically-located windings, when the Rogowski factor can be applied to correct the length of the magnetic path of stray flux in the air window. This is a modification of 1D analysis, which uses the discrete differential operator developed in [25,26]. Difficulties in formulating the boundary conditions for the case in which a magnetic field is generated by only one winding has been omitted.…”

Archives of Electrical Engineering

“…The FDOs of periodic and two-periodic time functions are presented in [14] and [15] and have been successfully tested in [16,17] for steady-state analysis of electromagnetic circuits. The same methodology has been used in [18] to develop FDOs to solve boundary-value problems for Ordinary Differential Equations (ODEs). The FDOs for 2D problems are shown in [19] and successfully tested in [20].…”

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…These operators re-lated the values of the derivatives at each point to the values of the function, at all points distributed uniformly over the function domain. The same types of FDOs were adapted to solve the one-dimensional (1D) boundary-value problems of ordinary differential equations [17,18] if the solution in the function domain repeated outside, i.e. it is periodic.…”