Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition
It is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of th...
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Belarusian National Technical University
2021-04-01
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| Series: | Наука и техника |
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| Online Access: | https://sat.bntu.by/jour/article/view/2432 |
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| author | I. N. Meleshko P. G. Lasy |
| author_facet | I. N. Meleshko P. G. Lasy |
| author_sort | I. N. Meleshko |
| collection | DOAJ |
| description | It is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of the Dirichlet integral in the unit disc centered at the origin on some admissible set of functions for given values of the normal derivative on the circle is equivalent to the Neimann boundary value problem for the Laplace equation in this domain. An effective approximate dilogarithm representation of the solution of the above equivalent variational boundary value problem is constructed on the basis of the known exact solution of the Neumann Boundary value problem for a circle using a special approximate formula for the Dini integral. The approximate formula is effective in the sense that it is quite simple in numerical implementation, stable, and the error estimation, which is uniform over a circle, allows calculations with the given accuracy. A special quadrature formula for the Dini integral has a remarkable property – its coefficients are non-negative. Quadrature formulas with non-negative coefficients occupy a special place in the theory of approximate calculations of definite integrals and its applications. Naturally, this property becomes even more significant when the coefficient are not number, but some functions. The performed numerical analysis of the approximate solution confirms its effectiveness. |
| format | Article |
| id | doaj-art-e41d6354db784be397f1a3e1f494c2e9 |
| institution | OA Journals |
| issn | 2227-1031 2414-0392 |
| language | Russian |
| publishDate | 2021-04-01 |
| publisher | Belarusian National Technical University |
| record_format | Article |
| series | Наука и техника |
| spelling | doaj-art-e41d6354db784be397f1a3e1f494c2e92025-08-20T02:36:06ZrusBelarusian National Technical UniversityНаука и техника2227-10312414-03922021-04-0120216817210.21122/2227-1031-2021-20-2-168-1722118Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary ConditionI. N. Meleshko0P. G. Lasy1Belarusian National Technical UniversityBelarusian National Technical UniversityIt is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of the Dirichlet integral in the unit disc centered at the origin on some admissible set of functions for given values of the normal derivative on the circle is equivalent to the Neimann boundary value problem for the Laplace equation in this domain. An effective approximate dilogarithm representation of the solution of the above equivalent variational boundary value problem is constructed on the basis of the known exact solution of the Neumann Boundary value problem for a circle using a special approximate formula for the Dini integral. The approximate formula is effective in the sense that it is quite simple in numerical implementation, stable, and the error estimation, which is uniform over a circle, allows calculations with the given accuracy. A special quadrature formula for the Dini integral has a remarkable property – its coefficients are non-negative. Quadrature formulas with non-negative coefficients occupy a special place in the theory of approximate calculations of definite integrals and its applications. Naturally, this property becomes even more significant when the coefficient are not number, but some functions. The performed numerical analysis of the approximate solution confirms its effectiveness.https://sat.bntu.by/jour/article/view/2432variational problemboundary neumann problemdini integralapproximate solutionquadrature formula |
| spellingShingle | I. N. Meleshko P. G. Lasy Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition Наука и техника variational problem boundary neumann problem dini integral approximate solution quadrature formula |
| title | Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition |
| title_full | Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition |
| title_fullStr | Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition |
| title_full_unstemmed | Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition |
| title_short | Approximate Dilogarithm Representation of One Variational Boundary Value Problem Solution for Circle under the Neumann Boundary Condition |
| title_sort | approximate dilogarithm representation of one variational boundary value problem solution for circle under the neumann boundary condition |
| topic | variational problem boundary neumann problem dini integral approximate solution quadrature formula |
| url | https://sat.bntu.by/jour/article/view/2432 |
| work_keys_str_mv | AT inmeleshko approximatedilogarithmrepresentationofonevariationalboundaryvalueproblemsolutionforcircleundertheneumannboundarycondition AT pglasy approximatedilogarithmrepresentationofonevariationalboundaryvalueproblemsolutionforcircleundertheneumannboundarycondition |