Application of generalized method of Lie algebraic discrete approximations for solving Cauchy problem with heat transfer equation (in Ukrainian)

Author
A. A. Kindybaliuk
Ivan Franko National University of Lviv
Abstract
Approximation properties and conditions of convergence of numerical scheme for solving Cauchy problem with heat transfer equation by means of generalized method of Lie-algebraic discrete approximations have been proved. Reduction of the Cauchy problem into system of linear algebraic equations provides power rates of convergence by all variables in equation.
Keywords
method of Lie-algebraic discrete approximations; power convergence; heat transfer equation
Reference
1. I.S. Berezin, N.P. Zhydkov, Numerical methods, V.1, M: Fismathyz, 1962. (in Russian)

2. O. Bihun, M. Prytula, The method of Lie algebraic discrete approximations in the theory of dynamical systems, Mat. Visnyk NTSH, 1 (2004), 2431. (in Ukrainian)

3. A.A. Kindybaliuk, M.M. Prytula, Generalization of the scheme of the Lie algebraic discrete approximations for Cauchy problem, XIX National ukrainian scientific conferece: Modern problems of applied mathematics and informatics. Theses. Lviv, (2013), 7374. (in Ukrainian)

4. L.A. Liusternik, V.I. Sobolev, Elements of functional analysis, .: Nauka, 1965. (in Russian)

5. M. Lustyk, A. Prykarpatski, M. Prytula, M. Vovk, Functional-operator analysis of converegence problems for F.Calogeros method of discrete approximations in Banach spaces, Math. Visnyk NTSH, 9 (2012), 168179. (in Ukrainian)

6. Ya A. Mytropolski, A.K. Prykarpatski, V.Hr. Samoylenko, Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical phisics, Ukr. Mat. Journ., 40 (1988), 453458. (in Russian)

7. R. Richtmyer, Difference methods for solving boundary-value problems, .: Mir, 1972. (in Russian)

8. .. Samarskii, .V. Gulin, Numerical methods: Handbook for students, .: Nauka, 1989. (in Russian)

9. V.Hr. Samoylenko, Algebraic scheme of discrete approximations for dynamical systems of mathematical phisics and estimations of its precision, Asymptotic methods in math-phisics problems .: Mathematical institute N USSR, (1988), 144151. (in Russian)

10. V.A. Trenogin, Functional analysis, : FIZMATLIT, 2002. (in Russian)

11. O.H. Bihun, Approximation properties of the Lie-algebraic scheme, Mat. Stud., 20 (2003), 1, 8591.

12. O.H. Bihun Modification of the Lie-algebraic scheme and approximation error estimations, Mat. Stud., 20 (2003), 2, 179184.

13. O.H. Bihun, M. Lustyk, Numerical tests and theoretical estimations for a Lie-algebraic scheme of discrete approximations, Visnyk of the Lviv University. Series of Applied Mathematics and Computer Science, 6 (2003), 310.

14. O. Bihun, M. Prytula, The rank of projection-algebraic representations of some differential operators, Mat. Stud. 35 (2011), 1, 921.

15. F. Calogero, Interpolation, differentiation and solution of eigenvalue problems in more than one dimension, Lett. Nuovo Cimento, 38 (1983), 13, 453459.

16. F. Calogero, E. Franko, Numerical tests of a novel technique to compute the eigen values of differential operators, Il Nuovo Cins., 89 (1985), 2, 161208.

17. F. Casas, Solution of linear partial differential equations by Lie algebraic method, Journ. of Comp. Appl. Math., 76 (1996), 159170.

18. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1990.

19. M. Lustyk, Lie-algebraic discrete approximation for nonlinear evolution equations, Journ. of Mathem. Sc., 109 (2002), 1, 11691172. 20. M. Lustyk, The Lie-algebraic discrete approximation scheme for evolution equations with Dirichlet/ Neumann data, Universitatis Iagellonicae Acta Mathematica, 40 (2002), 117-124.

21. A.K. Prykarpatsky, M.M. Prytula, O.O. Yerchenko, The Lie-algebraic discrete approximations in computing analysis, Volyn Mathematical Bulletin, 3 (1996), 113116.

22. J. Wei, E. Norman, On global representations of the solution of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327334.

23. F. Wolf, Lie algebraic solutions of linear Foker-Plank equations, Journ. Math. Phys., 29 (1988), 305307.

Pages
181-194
Volume
42
Issue
2
Year
2014
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue