Application of discrete differential operators of periodic functions to solve 1D boundary-value problems. Issue 4 (9th July 2020) - Sobczyk, Tadeusz; Jaraczewski, Marcin
Abstract : Purpose: 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. Design/methodology/approach: 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. Findings: An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems. 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 can be extended to the 2D or 3D cases with more flexible meshes. Practical implications: Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established. Originality/value: New finite-difference expressions for the first- and second-order derivatives have been applied.