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A note on the fractional hyperbolic differential and difference equations. (English) Zbl 1221.65212
The paper presents a first order difference scheme applied on an hyperbolic boundary value problem with a fractional differential equation with a self-adjoint operator A(t) formulated in a Hilbert space H. The stability estimates for the solution of the difference scheme are shown.
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35L10Second order hyperbolic equations, general
35R11Fractional partial differential 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]Lavoie, J. L.; Osler, T. J.; Tremblay, R.: Fractional derivatives and special functions, SIAM review 18, No. 2, 240-268 (1976) · Zbl 0324.44002 · doi:10.1137/1018042
[4]Tarasov, V. E.: Fractional derivative as fractional power of derivative, Int. J. Math. 18, 281-299 (2007) · Zbl 1119.26011 · doi:10.1142/S0129167X07004102
[5]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
[6]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
[7]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
[8]Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, 223-276 (1997)
[9]D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application 2, Lille, France, 1996.
[10]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
[11]Podlubny, I.; El-Sayed, A. M. A.: On two definitions of fractional calculus, Solvak acad. Sci.-inst. Exp. phys. (1996)
[12]H.O. Fattorini, Second Order Linear Differential Equations in Banach Space, Notas de Matematica, North-Holland, 1985.
[13]Piskarev, S.; Shaw, Y.: On certain operator families related to cosine operator function, Taiwanese J. Math. 1, No. 4, 3585-3592 (1997) · Zbl 0906.47030
[14]P.E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdat. Voronezh. Gosud. Univ., Voronezh, 1975, (in Russian).
[15]Krein, S. G.: Linear differential equations in a Banach space, (1966)
[16]Sobolevskii, P. E.; Pogorelenko, V. A.: Hyperbolic equations in a Hilbert space, Sibirskii matematichesskii zhurnal 8, No. 1, 123-145 (1967)
[17]Sobolevskii, P. E.; Chebotaryeva, L. M.: Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations, Izv. vyssh. Uchebn. zav., matematika 5, 103-116 (1977)
[18]A. Ashyralyev, M. Martinez, J. Paster, S. Piskarev, Weak maximal regularity for abstract hyperbolic problems in function spaces, Further progress in analysis, in: Proceedings of the Sixth International ISAAC Congress Ankara, Turkey 13 – 18 August 2007, World Scientific, 2009, pp. 679 – 689.
[19]Ashyralyev, A.; Aggez, N.: A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations, Numer. funct. Anal. optim. 25, No. 5 – 6, 1-24 (2004) · Zbl 1065.35021 · doi:10.1081/NFA-200041711
[20]A. Ashyralyev, I. Muradov, On difference schemes a second order of accuracy for hyperbolic equations, in: Modelling Processes of Explotation of Gas Places and Applied Problems of Theoretical Gasohydrodynamics, Ashgabat, Ilim, 1998, pp.127 – 138, (in Russian).
[21]Ashyralyev, A.; Sobolevskii, P. E.: New difference schemes for partial differential equations, Operator theor.: adv. Appl. 14 (2004)
[22]Ashyralyev, A.; Ozdemir, Y.: On nonlocal boundary value problems for hyperbolic-parabolic equations, Taiwanese J. Math. 11, No. 3, 1077-1091 (2007) · Zbl 1140.65039
[23]Ashyralyev, A.; Yildirim, O.: On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations, Taiwanese J. Math. 13, 22 (2009)
[24]Samarskii, A. A.; Gavrilyuk, I. P.; Makarov, V. L.: Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces, SIAM J numer. Anal. 39, No. 2, 708-723 (2001) · Zbl 1002.65101 · doi:10.1137/S0036142999357221
[25]Ashyralyev, A.; Sobolevskii, P. E.: Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations, Discrete dyn. Nature soc. 2005, No. 2, 183-213 (2005) · Zbl 1094.65077 · doi:10.1155/DDNS.2005.183
[26]Ashyralyev, A.; Koksal, M. E.: On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space, Numer. funct. Anal. optim. 26, No. 7 – 8, 739-772 (2005) · Zbl 1098.65055 · doi:10.1080/01630560500431068
[27]A. Ashyralyev, M.E. Koksal, On the stability of the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space, Discrete Dynamics in Nature and Society 2007, Article ID 57491, 2007, pp. 1 – 26. · Zbl 1156.65079 · doi:10.1155/2007/57491
[28]Ashyraliyev, M.: A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space, Numer. funct. Anal. optim. 29, No. 7-8, 750-769 (2008) · Zbl 1146.45001 · doi:10.1080/01630560802292069
[29]Ashyralyev, A.; Sobolevskii, P. E.: A note on the difference schemes for hyperbolic equations, Abstr. appl. Anal. 6, No. 2, 63-70 (2001) · Zbl 1007.65064 · doi:10.1155/S1085337501000501