Mathematical Methods in Engineering pp 13-22 | Cite as

# Fractional spaces generated by the positive differential and difference operators in a Banach space

Conference paper

## Abstract

The structure of the fractional spaces
with

*E*_{α,q},(*L*_{q}[0, 1],*A*^{x}) generated by the positive differential operator*A*^{x}defined by the formula*A*^{x}*u*= −*a*(*x*)*d*^{2}*u*/*dx*^{2}+*δu*, with domain*D*(*A*^{x}) = {*u*∈*C*^{(2)}[0, 1] :*u*(0) =*u*(1),*u*′(0) =*u*′(1)} is investigated. It is established that for any 0 <*α*< 1/2 the norms in the spaces*E*_{α,q}(*L*_{q}[0, 1],*A*^{x}) and*W*_{q}^{2α }[0, 1] are equivalent. The positivity of the differential operator*A*^{x}in*W*_{q}^{2α}[0, 1](0 ≤*α*< 1/2) is established. The discrete analogy of these results for the positive difference operator*A*_{ h }^{x}a second order of approximation of the differential operator*A*^{x}, defined by the formula$$
A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }}
{{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1
$$

*u*_{0}=*u*_{M}and −*u*_{2}+ 4*u*_{1}− 3*u*_{0}=*u*_{ M−2}− 4*u*_{ M−1}+ 3*u*_{M}is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.## Keywords

Banach Space Positive Operator Discrete Analogy Fractional Space Nonlocal Boundary Condition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [Kre66]Krein, S.G.: Linear Differential Equations in a Banach Space. Nauka, Moscow (1966) (Russian); English transl.: Linear Differential Equations in Banach space, Translations of Mathematical Monographs. Vol.23, American Mathematical Society, Providence RI (1968)Google Scholar
- [Gri84]Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Patman Adv. Publ. Program, London (1984)Google Scholar
- [Fat85]Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. Mathematics Studies, North-Holland (1985)MATHGoogle Scholar
- [Sol59]Solomyak, M.Z.: Analytic semigroups generated by elliptic operator in space
*L*_{p}. Dokl. Acad. Nauk SSSR,**127(1)**, 37–39 (1959) (Russian)MathSciNetGoogle Scholar - [Sol60]Solomyak, M.Z.: Estimation of norm of the resolvent of elliptic operator in spaces
*L*_{p}. Usp. Mat. Nauk,**15(6)**, 141–148 (1960) (Russian)MathSciNetGoogle Scholar - [KZPS76]Krasnosel’skii, M.A., Zabreiko, P.P., Pustyl’nik, E.I., Sobolevkii, P.E.: Integral Operators in Spaces of Summable Functions. Nauka, Moscow (1966) (Russian); English transl.: Integral Operators in Spaces of Summable Functions. Noordhoff, Leiden (1976).MATHGoogle Scholar
- [Ste80]Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Amer. Math. Soc.,
**259**, 299–310 (1980)MATHCrossRefMathSciNetGoogle Scholar - [AS94]Ashyralyev, A., Sobolevskii, P. E.: Well-Posedness of Parabolic Difference Equations. Birkhäuser Verlag, Basel Boston Berlin (1994)Google Scholar
- [AS04]Ashyralyev, A., Sobolevskii P.E.: New Difference schemes for Partial Differential equations. Birkhäuser Verlag, Basel Boston Berlin (2004)MATHGoogle Scholar
- [Sob71]Sobolevskii, P.E.: The coercive solvability of difference equations. Dokl. Acad. Nauk SSSR,
**201(5)**, 1063–1066 (1971) (Russian)MathSciNetGoogle Scholar - [AS77]Alibekov, Kh.A., Sobolevskii, P.E.: Stability of difference schemes for parabolic equations. Dokl. Acad. Nauk SSSR,
**232(4)**, 737–740 (1977) (Russian)MathSciNetGoogle Scholar - [AS79]Alibekov, Kh.A., Sobolevskii, P.E.: Stability and convergence of difference schemes of a high order for parabolic differential equations. Ukrain.Mat.Zh.,
**31(6)**, 627–634 (1979) (Russian)MATHMathSciNetGoogle Scholar - [AS84]Ashyralyev, A., Sobolevskii, P. E.: The linear operator interpolation theory and the stability of the difference-schemes. Doklady Akademii Nauk SSSR,
**275(6)**, 1289–1291 (1984) (Russian)MathSciNetGoogle Scholar - [Ash92]Ashyralyev, A.: Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations. Doctor of Sciences Thesis, Ins. of Math. of Acad. Sci., Kiev (1992) (Russian)Google Scholar
- [SS81]Smirnitskii, Yu.A., Sobolevskii, P.E.: Positivity of multidimensional difference operators in the
*C*—norm. Usp. Mat. Nauk,**36(4)**, 202–203 (1981) (Russian)Google Scholar - [Smi82]Smirnitskii, Yu.A.: Fractional powers of elliptic difference operators. PhD Thesis, Voronezh State University, Voronezh (1983) (Russian)Google Scholar
- [Dan89]Danelich, S.I.: Fractional powers of positive difference operators. PhD Thesis, Voronezh State University, Voronezh (1989) (Russian)Google Scholar
- [AY98]Ashyralyev, A., Yakubov, A.: Structures of fractional spaces generating by the transport operator. In: Muradov, A.N. (ed) Modeling the Processes in Exploration of Gas Deposits and Applied Problems of Theoretical Gas Hydrodynamics. Ilym, Ashgabat (1998) (Russian)Google Scholar
- [AY06]Ashyralyev, A., Yaz N.: On structure of fractional spaces generated by positivity operators with the nonlocal boundary conditions. In: Agarwal, R.P. (ed) Proceedings of the Conference Differential and Difference Equations and Applications. Hindawi Publishing Corporation, USA (2006)Google Scholar
- [AK95]Ashyralyev, A., Karakaya, I.: The structure of fractional spaces generated by the positive operator. In: Ashyralyev, Ch. (ed) Abstracts of Conference of Young Scientists. Turkmen Agricultural University, Ashgabat (1995)Google Scholar
- [AK01]Ashyralyev, A., Kendirli B.: Positivity in Holder norms of one dimensional difference operators with nonlocal boundary conditions. In: Cheshankov, B.I., Todorov, M.D. (ed) Application of Mathematics in Engineering and Economics
**26**. Heron Press-Technical University of Sofia, Sofia (2001)Google Scholar - [AYA05]Ashyralyev, A., Yenial-Altay N.: Positivity of difference operators generated by the nonlocal boundary conditions. In: Akca, H., Boucherif, A., Covachev, V. (ed) Dynamical Systems and Applications. GBS Publishers and Distributors, India (2005)Google Scholar
- [Tri78]Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library Vol. 18, Amsterdam (1978)Google Scholar

## Copyright information

© Springer 2007