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Well-posedness of the Basset problem in spaces of smooth functions. (English) Zbl 1217.34006

Summary: We consider the initial value problem for the fractional differential equation

du(t) dt+D t 1 2 u(t)+Au(t)=f(t),0<t<1,u(0)=0

in a Banach space E with the strongly positive operator A. The well-posedness of this problem in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for 2m-th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in the space variable are obtained.

34A08Fractional differential equations
34G10Linear ODE in abstract spaces
35K90Abstract parabolic equations
[1]Podlubny, I.: Fractional differential equations, (1999)
[2]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, (1993) · Zbl 0818.26003
[3]Kilbas, A. A.; Sristava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, Mathematics studies (2006)
[4]Lavoie, J. L.; Osler, T. J.; Tremblay, R.: Fractional derivatives and special functions, SIAM rev. 18, No. 2, 240-268 (1976) · Zbl 0324.44002 · doi:10.1137/1018042
[5]Tarasov, V. E.: Fractional derivative as fractional power of derivative, Int. J. Math. anal. 18, 281-299 (2007) · Zbl 1119.26011 · doi:10.1142/S0129167X07004102
[6]El-Mesiry, E. M.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. Comput. 160, No. 3, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[7]El-Sayed, A. M. A.; Gaafar, F. M.: Fractional order differential equations with memory and fractional-order relaxation-oscillation model, Pure math. Appl. 12 (2001) · Zbl 1006.34008
[8]El-Sayed, A. M. A.; El-Mesiry, E. M.; El-Saka, H. A. A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. appl. Math. 23, No. 1, 33-54 (2004) · Zbl 1213.34025 · doi:10.1590/S0101-82052004000100002 · doi:http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022004000100002&lng=en&nrm=iso&tlng=en
[9]Basset, A. B.: On the descent of a sphere in a viscous liquid, Q. J. Math. 42, 369-381 (1910) · Zbl 41.0826.01
[10]Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[11]Momani, Shaher; Al-Khaled, Kamel: Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. math. Comput. 162, No. 3, 1351-1365 (2005) · Zbl 1063.65055 · doi:10.1016/j.amc.2004.03.014
[12]Bagley, R. L.; Torvik, P. J.: On the appearance of the fractional derivative in the behavior of real materials, J. appl. Mech. 51, 294-298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615
[13]D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application 2, Lille, France, (1996).
[14]Ashyralyev, A.; A; Dal, F.; Pinar, Z.: On the numerical solution of fractional hyperbolic partial differential equations, Math. probl. Eng. 2009 (2009) · Zbl 1184.65083 · doi:10.1155/2009/730465
[15]Ashyralyev, A.: A note on fractional derivatives and fractional powers of operators, J. math. Anal. appl. 357, No. 1, 232-236 (2009) · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[16]Ashyralyev, A.; Dal, F.; Pinar, Z.: A note on the fractional hyperbolic differential and difference equations, Appl. math. Comput. 217, No. 9, 4654-4664 (2011) · Zbl 1221.65212 · doi:10.1016/j.amc.2010.11.017
[17]Dal, F.: Application of variational iteration method to fractional hyperbolic partial differential equations, Math. probl. Eng. 2009 (2009) · Zbl 1190.65185 · doi:10.1155/2009/824385
[18]Podlubny, I.; El-Sayed, A. M. A.: On two definitions of fractional calculus, Slovak acad. Sci., kosice, 21 (1996)
[19]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) · Zbl 0928.34042
[20]Agarwal, R.; Bohner, M.; Shakhmurov, V. B.: Maximal regular boundary value problems in Banach-valued weighted spaces, Bound. value probl. 1, 9-42 (2005) · Zbl 1081.35129 · doi:10.1155/BVP.2005.9
[21]Shakhmurov, V. B.: Coercive boundary value problems for regular degenerate differential-operator equations, J. math. Anal. appl. 292, No. 2, 605-620 (2004) · Zbl 1060.35045 · doi:10.1016/j.jmaa.2003.12.032
[22]Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems, Operator theory advances and applications (1995)
[23]Ashyralyev, A.; Hanalyev, A.; Sobolevskii, P. E.: Coercive solvability of nonlocal boundary value problem for parabolic equations, Abstr. appl. Anal. 6, No. 1, 53-61 (2001) · Zbl 0996.35027 · doi:10.1155/S1085337501000495
[24]Sobolevskii, P. E.: The coercive solvability of difference equations, Dokl. akad. Nauk SSSR 201, No. 5, 1063-1066 (1971) · Zbl 0246.39002
[25]P.E. Sobolevskii, Some properties of the solutions of differential equations in fractional spaces, in: Trudy Nauchn. -Issled. Inst. Mat. Voronezh. Gos. Univ. No. 14(1975), 68–74 (Russian).
[26]Ashyralyev, A.; Sobolevskii, P. E.: New difference schemes for partial differential equations, Operator theory: advances and applications 148 (2004)
[27]Ashyralyev, A.; Sobolevskii, P. E.: Well-posedness of parabolic difference equations, Operator theory advances and applications (1994)
[28]Smirnitskii, Yu.A.; Sobolevskii, P. E.: Positivity of multidimensional difference operators in the C-norm, Uspekhi. mat. Nauk 36, No. 4, 202-203 (1981)
[29]Yu.A. Smirnitskii, Fractional powers of elliptic difference operators. Ph.D. Thesis, Voronezh State University, Voronezh, 1983, Berlin, 2004 (in Russian).
[30]Ashyralyev, A.: Fractional spaces generated by the positive differential and difference operators in a Banach space, Proceedings of the conference ”mathematical methods and engineering”, 13-22 (2007) · Zbl 1130.46303 · doi:10.1007/978-1-4020-5678-9_2