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

Author 
a.kindybaluk@mail.ru
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 Liealgebraic
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 Liealgebraic 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), 24–31. (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. L’viv, (2013), 73–74. (in Ukrainian) 4. L.A. Liusternik, V.I. Sobolev, Elements of functional analysis, Ì.: Nauka, 1965. (in Russian) 5. M. Lustyk, A. Prykarpatski, M. Prytula, M. Vovk, Functionaloperator analysis of converegence problems for F.Calogero’s method of discrete approximations in Banach spaces, Math. Visnyk NTSH, 9 (2012), 168–179. (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), 453–458. (in Russian) 7. R. Richtmyer, Difference methods for solving boundaryvalue problems, Ì.: Mir, 1972. (in Russian) 8. À.À. Samarskii, À.V. Gulin, Numerical methods: Handbook for students, Ì.: 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 mathphisics problems Ê.: Mathematical institute ÀN USSR, (1988), 144–151. (in Russian) 10. V.A. Trenogin, Functional analysis, Ì: FIZMATLIT, 2002. (in Russian) 11. O.H. Bihun, Approximation properties of the Liealgebraic scheme, Mat. Stud., 20 (2003), ¹1, 85–91. 12. O.H. Bihun Modification of the Liealgebraic scheme and approximation error estimations, Mat. Stud., 20 (2003), ¹2, 179–184. 13. O.H. Bihun, M. Lustyk, Numerical tests and theoretical estimations for a Liealgebraic scheme of discrete approximations, Visnyk of the Lviv University. Series of Applied Mathematics and Computer Science, 6 (2003), 3–10. 14. O. Bihun, M. Prytula, The rank of projectionalgebraic representations of some differential operators, Mat. Stud. 35 (2011), ¹1, 9–21. 15. F. Calogero, Interpolation, differentiation and solution of eigenvalue problems in more than one dimension, Lett. Nuovo Cimento, 38 (1983), ¹13, 453–459. 16. F. Calogero, E. Franko, Numerical tests of a novel technique to compute the eigen values of differential operators, Il Nuovo Cins., 89 (1985), ¹2, 161–208. 17. F. Casas, Solution of linear partial differential equations by Lie algebraic method, Journ. of Comp. Appl. Math., 76 (1996), 159–170. 18. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1990. 19. M. Lustyk, Liealgebraic discrete approximation for nonlinear evolution equations, Journ. of Mathem. Sc., 109 (2002), ¹1, 1169–1172. 20. M. Lustyk, The Liealgebraic discrete approximation scheme for evolution equations with Dirichlet/ Neumann data, Universitatis Iagellonicae Acta Mathematica, 40 (2002), 117124. 21. A.K. Prykarpatsky, M.M. Prytula, O.O. Yerchenko, The Liealgebraic discrete approximations in computing analysis, Volyn Mathematical Bulletin, 3 (1996), 113–116. 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), 327–334. 23. F. Wolf, Lie algebraic solutions of linear FokerPlank equations, Journ. Math. Phys., 29 (1988), 305–307. 
Pages 
181194

Volume 
42

Issue 
2

Year 
2014

Journal 
Matematychni Studii

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