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High degree precision decomposition methodfor the evolution problem with an operator undera split form

Published online by Cambridge University Press:  15 September 2002

Zurab Gegechkori ,
Jemal Rogava  and
Mikheil Tsiklauri
Zurab Gegechkori
Affiliation:
Iv. Javakhishvili Tbilisi State University, Tbilisi 380043, Georgia. [email protected].
Jemal Rogava
Affiliation:
I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, Tbilisi 380043, Georgia. [email protected]. [email protected].
Mikheil Tsiklauri
Affiliation:
I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, Tbilisi 380043, Georgia. [email protected]. [email protected].
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Abstract

In the present work the symmetrized sequential-parallel decomposition methodof the third degree precision for the solution of Cauchy abstract problemwith an operator under a split form, is presented. The third degreeprecision is reached by introducing a complex coefficient with the positivereal part. For the considered schema the explicit a priori estimation isobtained.

Keywords

Decomposition methodSemigroupTrotterformulaCauchy abstract problem.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 693 - 704
Copyright
© EDP Sciences, SMAI, 2002

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References

Chernoff, P.R., Note on product formulas for operators semigroups. J. Funct. Anal. 2 (1968) 238-242. CrossRef
Chernoff, P.R., Semigroup product formulas and addition of unbounded operators. Bull. Amer. Mat. Soc. 76 (1970) 395-398. CrossRef
Dia, B.O. and Schatzman, M., Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées. RAIRO Modél. Math. Anal. Numér. 30 (1996) 343-383. CrossRef
Diakonov, E.G., Difference schemas with decomposition operator for Multidimensional problems. JNM and MPh 2 (1962) 311-319.
Fryazinov, I.V., Increased precision order economical schemas for the solution of parabolic type multidimensional equations. JNM and MPh 9 (1969) 1319-1326.
Z.G. Gegechkori, J.A. Rogava and M.A Tsiklauri, Sequential-Parallel method of high degree precision for Cauchy abstract problem solution. Tbilisi, in Reports of the Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 14 (1999).
Gordeziani, D.G., On application of local one dimensional method for solving parabolic type multidimensional problems of 2m-degree. Proc. Acad. Sci. GSSR 3 (1965) 535-542.
Gordeziani, D.G. and Meladze, H.V., On modeling multidimensional quasi-linear equation of parabolic type by one-dimensional ones. Proc. Acad. Sci. GSSR 60 (1970) 537-540.
Gordeziani, D.G. and Meladze, H.V., On modeling of third boundary value problem for the multidimensional parabolic equations of an arbitrary area by one-dimensional equations. JNM and MPh 14 (1974) 246-250.
D.G. Gordeziani and A.A. Samarskii, Some problems of plates and shells thermo elasticity and method of summary approximation. Complex Anal. Appl. (1978) 173-186.
N.N. Ianenko, Fractional steps method of solving multidimensional problems of mathematical physics. Nauka, Novosibirsk (1967) 196 p.
Ichinose, T. and Takanobu, S., The norm estimate of the difference between the Kac operator and the Schrodinger semigroup. Nagoya Math. J. 149 (1998) 53-81. CrossRef
K. Iosida, Functional analysis. Springer-Verlag (1965).
T. Kato, The theory of perturbations of linear operators. Mir, Moscow (1972) 740 p.
S.G. Krein, Linear equations in Banach space. Nauka, Moscow (1971), 464 p.
Kuzyk, A.M. and Makarov, V.L., Estimation of exactitude of summarized approximation of a solution of the Cauchy abstract problem. RAN USSR 275 (1984) 297-301.
G.I. Marchuk, Split methods. Nauka, Moscow (1988) 264 p.
Rogava, J.A., On the error estimation of Trotter type formulas in the case of self-Adjoint operator. Funct. Anal. Appl. 27 (1993) 84-86. CrossRef
J.A. Rogava, Semi-discrete schemas for operator differential equations. Tbilisi, Georgian Technical University press (1995) 288 p.
A.A. Samarskii, Difference schemas theory. Nauka, Moscow (1977), 656 p.
A.A. Samarskii and P.N. Vabishchevich, Additive schemas for mathematical physics problems. Nauka, Moscow (1999).
R. Temam, Quelques méthodes de décomposition en analyse numérique. Actes Congrés Intern. Math. (1970) 311-319.
Trotter, H., On the product of semigroup of operators. Proc. Amer. Mat. Soc. 10 (1959) 545-551. CrossRef