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Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2017, Number 50, Pages 9–29
DOI: https://doi.org/10.17223/19988621/50/2
(Mi vtgu615)
 

This article is cited in 7 scientific papers (total in 7 papers)

MATHEMATICS

On solving plane problems of non-stationary heat conduction by the collocation boundary element method

D. Yu. Ivanov

Moscow State University of Railway Engeneering (MIIT), Moscow, Russian Federation
Full-text PDF (622 kB) Citations (7)
References:
Abstract: In this paper, we propose a fully justified collocation boundary element method allowing one to obtain numerical solutions of internal and external initial-boundary value problems (IBVPs) with boundary conditions of the first, second, and third kind for the equation tu=a2Δ2uputu=a2Δ2upu with constants a,p>0a,p>0 in a plane spatial domain ΩΩ (in a bounded one Ω+Ω+ or in its exterior ΩΩ) on a finite time interval It[0,T]It[0,T] at a zero initial condition. The solutions are found in the form of the double-layer potential for the Dirichlet IBVP and in the form of the simple layer potential for the Neumann–Robin IBVP with unknown density functions determined from the boundary integral equations (BIEs) of the second kind.
In this paper, instead of the usual piecewise-polynomial interpolation of the density function on time variable ττ, the BIEs are approximated by the piecewise-quadratic interpolation (PQI) of the C0C0-semigroup of right shifts on time. Also, on the basis of the PQI, the approximation of the multiplier epτepτ in kernels of the integral operators is carried out. In addition, the PQI of density functions is performed: for the BIE, only on arc-length ss; for the potentials, on both variables ss and ττ. Then, the integration with respect to the variable ττ on the boundary elements (BEs) is performed exactly. The integration with respect to the variable ss on the BE for the potentials is performed approximately by using the Gaussian quadrature with γ2 points. For the BIE, the integration with respect to the arc-length s is carried out in two ways. On singular BEs and on nearby singular BEs, adjacent to a singular BE in some fixed arc-length region, an exact integration with respect to the variable r is carried out (r is the distance from the boundary point at which the integral is calculated as a function of parameter to the current boundary point of the integration). In this integration, functions of the variable r are taken as the weighting functions. The functions of r are generated by the fundamental solution of the heat equation and the rest of the integrand is approximated by quadratic interpolation on r. The integrals with respect to s on the remaining BEs are calculated using the Gaussian quadrature with γ points.
The cubic convergence of approximate solutions of the IBVP at any point of the set Ω×It is proved under conditions ΩC5C2γ and wC31,3(Ω). It is also proved that such solutions are resistant to perturbations of the boundary function w in the norm of the space C01(Ω). Here, Ckm,n(Ω)Ckm(Ω)C0m+n(Ω) and Ckm(Ω) is the Banach space of k times continuously differentiable on Ω vector functions with values in Sobolev's space which is the domain of definition of the operator Bm ((Bf)(t)=f(t), f(t=0)=0).
In conclusion, results of the numerical experiments are presented. They confirm the cubic convergence of approximate solutions for all three IBVPs in a circular domain.
Keywords: boundary integral equation, boundary element method, singular boundary elements, non-stationary heat conduction, collocation, operator, approximation, stability.
Received: 07.09.2017
Bibliographic databases:
Document Type: Article
UDC: 519.642.4
Language: Russian
Citation: D. Yu. Ivanov, “On solving plane problems of non-stationary heat conduction by the collocation boundary element method”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, no. 50, 9–29
Citation in format AMSBIB
\Bibitem{Iva17}
\by D.~Yu.~Ivanov
\paper On solving plane problems of non-stationary heat conduction by the collocation boundary element method
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2017
\issue 50
\pages 9--29
\mathnet{http://mi.mathnet.ru/vtgu615}
\crossref{https://doi.org/10.17223/19988621/50/2}
\elib{https://elibrary.ru/item.asp?id=30778968}
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  • https://www.mathnet.ru/eng/vtgu/y2017/i50/p9
  • This publication is cited in the following 7 articles:
    1. D. Yu. Ivanov, “On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain”, Ufa Math. J., 15:4 (2023), 76–99  mathnet  crossref
    2. Ivanov D.Yu., “Ob approksimatsii normalnoi proizvodnoi teplovogo potentsiala prostogo sloya vblizi granitsy dvumernoi oblasti”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2023, no. 83, 31–51  mathnet  crossref
    3. D. Yu. Ivanov, O. A. Platonova, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2526, MODERN APPROACHES IN ENGINEERING AND NATURAL SCIENCES: MAENS-2021, 2023, 020007  crossref
    4. Ivanov D.Yu., “O ravnomernoi skhodimosti approksimatsii potentsiala dvoinogo sloya vblizi granitsy dvumernoi oblasti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:1 (2022), 26–43  mathnet  crossref  mathscinet
    5. Ivanov D.Yu., “O sovmestnom primenenii kollokatsionnogo metoda granichnykh elementov i metoda Fure dlya resheniya zadach teploprovodnosti v konechnykh tsilindrakh s gladkimi napravlyayuschimi”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 72, 15–38  mathnet  crossref
    6. Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy dvumernoi oblasti s pomoschyu poluanaliticheskoi approksimatsii teplovogo potentsiala dvoinogo sloya”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2020, no. 65, 30–52  mathnet  crossref
    7. Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy oblasti v sluchae dvumernykh zadach nestatsionarnoi teploprovodnosti s granichnymi usloviyami vtorogo i tretego roda”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2019, no. 57, 5–25  mathnet  crossref  elib
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Томского государственного университета. Математика и механика
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