Well- posedness of the Rothe difference scheme for reverse parabolic equations

Iranian Journal of Optimization

مقاله 1، دوره 2، شماره 1، فروردین 2010، صفحه 77-99   

نوع مقاله: Research Paper

نویسندگان

Allaberen Ashyralyev¹؛ Ayfer Dural²؛ Yaşar Sözen^{* 1}

¹Department of Mathematics, Fatih University, Istanbul,34500, Turkey

²Gaziosman Paşa Lisesi Istanbul, Turkey

چکیده

 We consider the Rothe difference scheme for approximate solution of the abstract parabolic equation in a Hilbert space with the nonlocal boundary condition. Theorems on stability estimates, coercivity and almost coercivity estimates for the solution of this difference scheme are established. In application, new coercivity inequalities for the solution of multi-point nonlocal boundary value difference equations of parabolic type are obtained.

کلیدواژه‌ها

Multipoint nonlocal boundary value problem؛ parabolic equations؛ reverse type؛ difference equations؛ well-posedness؛ almost coercivity 2000 MSC: 47D06؛ 35K20

مراجع

[1] Aibeche A., Favvini A., Coerciveness estimate for Ventcel boundary value problem for a differential equation , Semigroup Forum, 70, No 2, 269-277, 2005.

 [2] Agarwal R., Bohner M., V.B. , Shakhmurov, Maximal regular boundary    value problems in Banach-valued weighted spaces , Bound Value Probl., 1, 9-42, 2005.

[3] Ashyralyev A., Nonlocal boundary value problems for partial differential equations:well-posedness , AIP Conference Proceedings Global Analysis and Applied Mathematics: International Workshop on Global Analysis, 729, 325-331, 2004.

[4] Ashyralyev A., Nonlocal boundary value problems for abstract parabolic difference equations: well-posedness in Bochner spaces , J. Evol. Equ., 6, No 1,  1-28, 2006.

[5] Ashyralyev A., Hanalyev A., Sobolevskii, P.E., Coercive solvability of non local boundary value problem for parabolic equations , Abstr. Appl. Anal., 6, No 1, 53-61, 2001.

[6] Ashyralyev A., Dural A., Sözen Y., Multipoint nonlocal boundary valueo problems for reverse parabolic equations: well-posedness , Vestnik of Odessa National University: Mathematics and Mechanics, 13, 1-12, 2008.

[7] Ashyralyev A., Sobolevskii P.E., Positive operators and the fractional spaces , Offset Laboratory VSU, Voronezh, Russia, 1989.

[8] Ashyralyev A.,  Sobolevskii P.E., Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkh¨user, Basel, Switzerland, 1994.

[9] Ashyralyev A., Sobolevskii P.E., New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkh¨user, Basel, Switzerland, 2004.

[10] Sobolevskii P.E., Coerciveness inequalities for abstract parabolic equations , Dokl. Akad. Nauk SSSR, 157, No 1, 52-55, 1964.

[11] Clement Ph., Guerre-Delabrire S., On the regularity of abstract Cauchy problems and boundary value problems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9, No 4,  245-266, 1999.

[12] Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N., Linear and Quasilinear Equations of Parabolic Type. Moscow: Nauka, 1967, English transl. Linear and Quasilinear Equations of Parabolic Type, Providence, RI: American Mathematical Society, Translations of Mathematical Monographs, 23, 1968.

[13] Ladyzhenskaya O.A., Ural’tseva N.N., Linear and Quasilinear Equations of Elliptic Type, Moscow: Nauka, 1973 , English transl. of first edition, Linear and Quasilinear Elliptic Equations, London, New York:  Academic Press, 1968.

[14] ShakhmurovV.B., Coercive boundary value problems for regular degenerate differential -operator equations, J. Math. Anal. Appl., 292, No 2, 605-620, 2004.

[15] Sobolevskii P.E., Some properties of the solutions of differential equations in fractional spaces in: Trudy Nauchn. Issled. Inst. Mat. Voronezh. Gos.  Univ., 14, 68-74, 1975.

[16] Sobolevskii P.E., The coercive solvability of difference equations, Dokl.     Acad. Nauk SSSR, 201-1971, No 5, 1063-1066, 1971.

[17] Vishik M.L., Myshkis A.D.,  Oleinik O.A., Partial Differential Equations in: Mathematics in USSR in the Last 40 Years, 1917-1957, Moscow: Fizmatgiz, 1,563-599, 1959.