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Abstract

This paper presents novel discrete differential operators for periodic functions of one- and two-variables, which relate the values of the derivatives to the values of the function itself for a set of arbitrarily chosen points over the function’s area. It is very characteristic, that the values of the derivatives at each point depend on the function values at all points in that area. Such operators allow one to easily create finite-difference equations for boundaryvalue problems. The operators are addressed especially to nonlinear differential equations.
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Bibliography

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[9] Demenko A., Sykulski J., On the equivalence of finite difference and edge element formulations in magnetic field analysis using vector potential, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 33, no. 1/2, pp. 47–55 (2014).
[10] Huang J., LiaoW., Li Z., A multi-block finite difference method for seismic wave equation in auxiliary coordinate system with irregular fluid–solid interface, Engineering Computations, vol. 35, no. 1, pp. 334–362 (2018).
[11] Chapwanya M., Dozva R., Gift Muchatibaya G., A nonstandard finite difference technique for singular Lane-Emden type equations, Engineering Computations, vol. 36, no. 5, pp. 1566–1578 (2019).
[12] Mawlood M., Basri S., AsrarW., Omar A., Mokhtar A., Ahmad M., Solution of Navier-Stokes equations by fourth-order compact schemes and AUSM flux splitting, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 16, no. 1, pp. 107–120 (2006).
[13] Ivanovic M., Svicevic M., Savovic S., Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 27, no. 12, pp. 2682–2695 (2017).
[14] Sobczyk T.J., Algorithm for determining two-periodic steady-states in AC machines directly in time domain, Archives of Electrical Engineering, Polish Academy of Science, Electrical Engineering Committee, vol. 65, no. 3, pp. 575–583 (2016), DOI: 10.1515/aee-2016-0041.
[15] Sobczyk T.J., Radzik M., Radwan-Pragłowska N., Discrete differential operators for periodic and two-periodic time functions, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 38, no. 1, pp. 325–347 (2019).
[16] Sobczyk T.J., Radzik M., A new approach to steady state analysis of nonlinear electrical circuits, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 37, no. 3, pp. 716–728 (2017).
[17] Sobczyk T.J., Radzik M., Tulicki J., Direct steady-state solutions for circuit models of nonlinear electromagnetic devices, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 40, no. 3, pp. 660–675 (2021), DOI: 10.1108/COMPEL-10-2020-0324.
[18] Sobczyk T.J., Jaraczewski M., Application of discrete differential operators of periodic functions to solve 1D boundary-value problems, COMPEL – The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald Pub. Ltd., vol. 39, no. 4, pp. 885–897 (2020).
[19] Sobczyk T.J., 2D discrete operators for periodic functions, Proceedings IEEE Conference Selected Issues of Electrical Engineering and Electronics (WZZE), Zakopane, Poland, pp. 1–5 (2019), https://ieeexplore.ieee.org/document/8979992.
[20] Jaraczewski M., Sobczyk T., Leakage Inductances of Transformers at Arbitrarily Located Windings, Energies, vol. 13, no. 23, 6464 (2020), DOI: 10.3390/en13236464.

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Authors and Affiliations

Tadeusz Jan Sobczyk
1
ORCID: ORCID

  1. Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Cracow University of Technology, 24 Warszawska str., 31-155 Kraków, Poland
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Abstract

This paper presents the concept of using algorithms for reducing the dimensions of finite-difference equations of two-dimensional (2D) problems, for second-order partial differential equations. Solutions are predicted as two-variable functions over the rectangular domain, which are periodic with respect to each variable and which repeat outside the domain. Novel finite-difference operators, of both the first and second orders, are developed for such functions. These operators relate the value of derivatives at each point to the values of the function at all points distributed uniformly over the function domain. A specific feature of the novel operators follows from the arrangement of the function values as well as the values of derivatives, which are rectangular matrices instead of vectors. This significantly reduces the dimensions of the finite-difference operators to the numbers of points in each direction of the 2D area. The finite-difference equations are created exemplary elliptic equations. An original iterative algorithm is proposed for reducing the process of solving finite-difference equations to the multiplication of matrices.

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Authors and Affiliations

T. Sobczyk

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