General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions
Zoltan Vizvari,
Mihaly Klincsik,
Peter Odry,
Vladimir Tadic,
Zoltan Sari
Affiliations
Zoltan Vizvari
Department of Environmental Engineering, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany str. 2, H-7624 Pecs, Hungary
Mihaly Klincsik
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
Peter Odry
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
Vladimir Tadic
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
Zoltan Sari
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
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.
