Calculation of the green’s function of boundary value problems for linear ordinary differential equations

Authors

DOI:

https://doi.org/10.15587/1729-4061.2020.193470

Keywords:

Green's function, ordinary differential equations, power series, generalized power series, boundary value problems

Abstract

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory.

The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Author Biographies

Irina Belyaeva, Belgorod National Research University Pobedy str., 85, Belgorod, Russia, 308015

PhD, Associate Professor

Department of Computer Science, Natural Sciences and Teaching Methods

Nikalay Chekanov, Belgorod National Research University Pobedy str., 85, Belgorod, Russia, 308015

Doctor of Physical and Mathematical Sciences, Professor

Department of Applied Mathematics and Computer Modeling

Natalia Chekanova, Kharkiv Educational and Scientific Institute of SHEI “Banking University” Peremohy ave., 55, Kharkiv, Ukraine, 61174

PhD, Associate Professor

Department of Information Technology

Igor Kirichenko, National University of Civil Defence of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023

Doctor of Physical and Mathematical Sciences, Professor

Department of Physical and Mathematical Sciences

Oleg Ptashny, Kharkiv National Automobile and Highway University Yaroslava Mudroho str., 25, Kharkiv, Ukraine, 61002

PhD, Associate Professor

Department of Higher Mathematics

Tetyana Yarkho, Kharkiv National Automobile and Highway University Yaroslava Mudroho str., 25, Kharkiv, Ukraine, 61002

Doctor of Pedagogical Sciences, Associate Professor, PhD, Head of Department

Department of Higher Mathematics

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Published

2020-02-29

How to Cite

Belyaeva, I., Chekanov, N., Chekanova, N., Kirichenko, I., Ptashny, O., & Yarkho, T. (2020). Calculation of the green’s function of boundary value problems for linear ordinary differential equations. Eastern-European Journal of Enterprise Technologies, 1(4 (103), 43–52. https://doi.org/10.15587/1729-4061.2020.193470

Issue

Vol. 1 No. 4 (103) (2020): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2020 Igor Kirichenko, Irina Belyaeva, Oleg Ptashny, Nikalay Chekanov, Natalia Chekanova, Tetyana Yarkho

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