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Error estimates for projection-difference methods for differential equations with differentiable operators

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Abstract

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to A(0).

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References

  1. P. E. Oya, “On the Convergence and Stability of the Galyorkin Method for Parabolic Equations with Differentiable Operators,” Uchen. Zap. Tartusk. Univ., No. 374, 194–210 (1975).

  2. G. M. Vainikko and P. E. Oya, “On Convergence and Rate of Convergence of the Galyorkin Method for Abstract Evolution Equations,” Differents. Uravneniya 11(7), 1269–1277 (1975).

    MATH  Google Scholar 

  3. S. G. Krein, Linear Differential Equations in Banach Spaces (Nauka, Moscow, 1967; American Mathematical Society, Providence, 1971).

    Google Scholar 

  4. J.-L. Lions, QuelquesMéthodes de Résolution des Problèmes aux Limites non Linéaires (Dunod, Paris, 1969; Mir,Moscow, 1972).

    Google Scholar 

  5. H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferential-Gleichungen (Akademie-Verlag, Berlin, 1974; Mir,Moscow, 1978).

    Google Scholar 

  6. H. O. Fattorini, The Cauchy Problem (Addison-Wesley Publishing Company, Massachusets, 1983).

    MATH  Google Scholar 

  7. P. E. Sobolevskii, “On Equations with Operators Forming an Acute Angle,” Sov. Phys. Dokl. 116(5), 754–757 (1957).

    MathSciNet  Google Scholar 

  8. V. V. Smagin, “Estimates of the Rate of Convergence of Projective and Projective-Difference Methods for Weakly Solvable Parabolic Equations,” Matem. Sborn. 188(3), 143–160 (1997).

    MathSciNet  Google Scholar 

  9. A. G. Zarubin, “On the Convergence Rate of the Faedo-Galyorkin Method for the Linear Nonstationary Equations,” Differents. Uravneniya 18(4), 639–645 (1982).

    MathSciNet  Google Scholar 

  10. A. G. Zarubin, “Rate of Convergence of the Faedo-Galyorkin Method for Quasilinear Nonstationary Operator Equations,” Differents. Uravneniya 26(12), 2051–2059 (1990).

    MathSciNet  Google Scholar 

  11. E. Hausenblas, “Numerical Analysis of Semilinear Stochastic Evolution Equations in Banach Spaces,” J. Comput. Appl.Math. 147(2), 485–516 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. D. Lyashko, “Projection-Difference Schemes for Hyperbolic Equations with a Higher-Order Degenerate Elliptic Operator,” Differents. Uravneniya 43(7), 972–977 (2007).

    MathSciNet  Google Scholar 

  13. H. Li and R. Liu, “The Space-Time Finite Element Method for Parabolic Problems,” Appl.Mathematics and Mechanics 22(6), 687–700 (2001).

    Article  MATH  Google Scholar 

  14. I. D. Turetaev, “Exact Estimates of the Error Gradient for Projection-Difference Schemes for Parabolic Equations in an Arbitrary Domain,” Zhurn. Vychisl. Matem. i Matem. Fiz. 26(11), 1748–1751 (1986).

    MathSciNet  Google Scholar 

  15. V. V. Smagin, “Energy Error Estimates for the Projection-Difference Method with the Crank-Nicolson Scheme for Parabolic Equations,” Sib. Matem. Zhurn. 42(3), 670–682 (2001).

    MATH  MathSciNet  Google Scholar 

  16. A. Ashyralyev and P. E. Sobolevskii, “Well-Posed Solvability of the Cauchy Problem for Difference Equation of Parabolic Type,” Nonlinear Anal. 24(2), 257–264 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Ashyralyev, S. Piskarev, and L. Weis, “On Well-Posedness of Difference Schemes for Abstract Parabolic Equations,” Numer. Funct. Anal. and Optimization 23(7–8), 669–693 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  18. S. G. Mikhlin, Variational Methods in Mathematical Physics (Nauka, Moscow, 1970).

    Google Scholar 

  19. A. G. Zarubin and M. F. Tiunchik, “On Rothe-Galyorkin Method for One Class of Linear Nonstationary Equations,” Differents. Uravneniya 19(12), 2141–2148 (1983).

    MathSciNet  Google Scholar 

  20. A. G. Zarubin, “The Rothe and Rothe-Galyorkin Methods for a Quasilinear Operator-Differential Equation,” Differents. Uravneniya 32(12), 1683–1690 (1996).

    MathSciNet  Google Scholar 

  21. P. V. Vinogradova, “Error Estimates for a Projection-Difference Method for a Linear Differential-Operator Equation,” Differents. Uravneniya 44(7), 942–951 (2008).

    MathSciNet  Google Scholar 

  22. V. B. Demidovich, “The Asymptotic Behaviour of the Solutions of Finite Difference Equations,” Differents. Uravneniya 10(12), 2267–2278 (1974).

    Google Scholar 

  23. A. G. Zarubin, “Investigation by the Method of Fractional Degrees of the Projection Solution Procedure for Linear and Quasilinear Equations,” Doctoral Dissertation in Mathematics and Physics (Khabarovsk, 1986).

  24. A. A. Samarskii and A. V. Gulin, Numerical Methods (Nauka, Moscow, 1989).

    MATH  Google Scholar 

  25. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Amer.Math. Soc., Providence, RI, 1968).

    MATH  Google Scholar 

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  1. Far Eastern State Transport University, ul. Serysheva 47, Khabarovsk, 680021, Russia

    P. V. Vinogradova

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  1. P. V. Vinogradova
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Correspondence to P. V. Vinogradova.

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Original Russian Text © P.V. Vinogradova, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 7, pp. 3–15.

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Vinogradova, P.V. Error estimates for projection-difference methods for differential equations with differentiable operators. Russ Math. 54, 1–11 (2010). https://doi.org/10.3103/S1066369X10070017

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