Error message

  • Deprecated function: Unparenthesized `a ? b : c ? d : e` is deprecated. Use either `(a ? b : c) ? d : e` or `a ? b : (c ? d : e)` in include_once() (line 1439 of /home/science2016/public_html/includes/bootstrap.inc).
  • Deprecated function: Array and string offset access syntax with curly braces is deprecated in include_once() (line 3557 of /home/science2016/public_html/includes/bootstrap.inc).

Pipeline pressure distribution finding methods

Pyanylo Ya., Sobko V.
AttachmentSize
PDF icon 2016_3_2_199_207.pdf197.26 KB

Keywords:

Abstract: 
The method of solving problems of mathematical physics, in particular for calculating a non-stationary gas flow in pipelines, is proposed in this article on the basis of the biorthogonal polynomial constructed by the authors. The method of solving the problem by means of the separation of variables in the base of biorthogonal polynomials is investigated. The analytical-approximate and approximate solutions of the problem as the sum of some biorthogonal and quasi-spectral polynomials are found. The comparative analysis between the obtained analytical-approximate and approximate solutions is conducted. The influence of parameters of methods, including the order of the partial sum, a bit grid, and an accuracy error of calculations on the obtained solution are studied. The results of calculation are presented in the form of tables.
References: 
  1. Аbramovich М., Stygan I. Reference book of Special Functions with Formulae, Diagrams and Mathematical Tables. Мoscow, Science (1979).
  2. Gletcher К. Numerical Methods of the Base of the Galierkin’s Method. Мoscow, World (1988).
  3. Dziadyk V. К. Аpproximate Methods of Solution of Differential and Integral Equations. Кyiv, Scientific Thought (1998).
  4. Dziadyk V. К. Introduction in to the Theory of Uniform Approximation of Functions by Polynomials. Мoscow, Science (1977).
  5. Каntorovich L. V., Аkilov G. P. Functional Analysis. Мoscow, Science (1984).
  6. Кorneychuk N. P. Accurate Constants in the Theory of Approximation. Мoscow, Science (1987).
  7. Lаntsosh К. Practical Methods of Applied Analysis. Moscow, State Publishing House of Physical-Mathematical Literaure (1961).
  8. Ditkin V. А., Prudnikov А. P. Оperational Calculation. Мoscow, Higher School (1975).
  9. Lopukh N., Pritula М., Pjanylo Ya., Savula Ya. Algorithms of Calculation of Hydrodynamical Parameters of Gas Flow in Pipelines. Visnyk of Lviv University. Serie of Appl. Math. and Comp. Sci, 12, 108–117 (2007) (in Ukrainian).
  10. Pyanylo Ya. D., Sobko V. G. Building and Research of Biorthogonal Polynomials on the Base of the Chebyshev Polynomials. Appl. Problems of Mech. and Math. 11, 135–141 (2013) (in Ukrainian).
  11. Pyanylo Ya., Sobko V. Research of Peculiarities of Spectral Schedules in the Bases of Orthogonal, Quaziorthogonal, Biorthogonal Polynomials. Phisical-Mathematical Modelling and Information Techologies. 19, 146–156 (2014) (in Ukrainian).
  12. Ghoreishi F., Mohammad Hosseini S. The Tau method and a new preconditioner. J. Comp. Appl. Math. 163(2), 351–379 (2004).
  13. Coutsias E. A., Hagstorm T., Hesthaven J. S., Torres D. Integration preconditioners for differential operators in spectral τ-methods. Proc. 3rd International Conference on Spectral and High Order Methods, Houston, TX, 21–38 (1995).
  14. Bernardi C., Maday Y. Properties of some weighted sobolev spaces, and applications to spectral approximations. SIAM J. Numer. Anal. 26, 769–829 (1989).
  15. Jie Shen. Efficient spectral-Galerkin method II. Direct solvers for second- and fourthorder equations by using Chebyshev polynomials. SIAM J. Sci. Comput. 16, 74–87 (1995).
  16. Jie Shen. Efficient Chebyshev-Legendre Galerkin methods for elliptic problems. In A. V. Ilin and R. Scott, editors, Proceedings of ICOSAHOM’95. Houston. J. Math. 233–240 (1996).
  17. Badkov V. M. Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval. Math. USSR Sbornik. 2, MR0355464 (50:7938) 223–256 (1974).
  18. John P. Boyd. Chebyshev and Fourier Spectral methods, 2nd edition. Dover Publication, Inc., Mineola, New York (2001).
  19. Walte A. Strauss. Partial Differential Equations: An Introduction, 2nd edition, John Wiley and Sons (2008).
  20. Cabos Ch. A preconditioning of the tau operator for ordinary differential equations. Z. Angew. Math. Mech. 74 (11), 521–532 (1994).
  21. Coutsias E. A., Hagstorm T., Hesthaven J., Torres D. Integration preconditioners for differential operators in spectral τ-methods. Proc. 3rd International Conference on Spectral and High Order Methods, Houston, TX, 21–38 (1995).
Bibliography: 
Math. Model. Comput. Vol.3, No.2, pp.199-207 (2016)