{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:31Z","timestamp":1760148991739,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,27]],"date-time":"2023-06-27T00:00:00Z","timestamp":1687824000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and\/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.<\/jats:p>","DOI":"10.3390\/axioms12070633","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T00:50:52Z","timestamp":1687913452000},"page":"633","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0313-9959","authenticated-orcid":false,"given":"Zoltan","family":"Vizvari","sequence":"first","affiliation":[{"name":"Department of Environmental Engineering, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 2, H-7624 Pecs, Hungary"},{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag str. 20, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2831-8750","authenticated-orcid":false,"given":"Mihaly","family":"Klincsik","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag str. 20, H-7624 Pecs, Hungary"},{"name":"Department of Technical Informatics, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Odry","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"},{"name":"Institute of Information Technology, University of Dunaujvaros, Tancsics M. str. 1\/A, H-2401 Dunaujvaros, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8796-6822","authenticated-orcid":false,"given":"Vladimir","family":"Tadic","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"},{"name":"Institute of Information Technology, University of Dunaujvaros, Tancsics M. str. 1\/A, H-2401 Dunaujvaros, Hungary"},{"name":"John von Neumann Faculty of Informatics, University of Obuda, Becsi str. 96\/B, H-1034 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zoltan","family":"Sari","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag str. 20, H-7624 Pecs, Hungary"},{"name":"Department of Technical Informatics, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 6, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114","DOI":"10.3126\/jjis.v5i0.17844","article-title":"Application of differential equation in L-R and C-R circuit analysis by classical method","volume":"5","author":"Regmi","year":"2017","journal-title":"Janapriya J. 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