Weighted Estimates for Boundary Value Problems with Fractional Derivatives

References

[1] A. Ashyralyev, A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl. 357 (2009), no. 1, 232–236. 10.1016/j.jmaa.2009.04.012Search in Google Scholar

[2] T. Y. Cheung, Three nonlinear initial value problems of the hyperbolic type, SIAM J. Numer. Anal. 14 (1977), no. 3, 484–491. 10.1137/0714028Search in Google Scholar

[3] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Math. 2004, Springer, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar

[4] K. Diethelm, N. J. Ford, A. D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 6–8, 743–773. 10.1016/j.cma.2004.06.006Search in Google Scholar

[5] B. Duan, R. Lazarov and J. Pasciak, Numerical approximation of fractional powers of elliptic operators, preprint (2018), https://arxiv.org/abs/1803.10055. 10.1093/imanum/drz013Search in Google Scholar

[6] N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal. 14 (2011), no. 4, 554–567. 10.2478/s13540-011-0034-4Search in Google Scholar

[7] R. A. Frisch and B. R. S. Cheo, On a bounded one-dimensional Poisson–Vlasov system, SIAM J. Appl. Math. 24 (1973), 362–368. 10.1137/0124038Search in Google Scholar

[8] Y. F. Galba, On the order of accuracy of the difference scheme for Poisson’s equation with a mixed boundary condition, Optimization of Algorithms of Computer Software, V. M. Glushkov Institute of Cybernetics of AN of UkrSSR, Kyiv (1985), 30–34. Search in Google Scholar

[9] I. P. Gavrilyuk, An algorithmic representation of fractional powers of positive operators, Numer. Funct. Anal. Optim. 17 (1996), no. 3–4, 293–305. 10.1080/01630569608816695Search in Google Scholar

[10] I. P. Gavrilyuk, Super exponentially convergent approximation to the solution of the Schrödinger equation in abstract setting, Comput. Methods Appl. Math. 10 (2010), no. 4, 345–358. 10.2478/cmam-2010-0020Search in Google Scholar

[11] I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij, Data-sparse approximation to a class of operator-valued functions, Math. Comp. 74 (2005), no. 250, 681–708. 10.1090/S0025-5718-04-01703-XSearch in Google Scholar

[12] I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij, Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing 74 (2005), no. 2, 131–157. 10.1007/s00607-004-0086-ySearch in Google Scholar

[13] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/3779Search in Google Scholar

[14] B. S. Jovanović, L. G. Vulkov and A. Delić, Boundary value problems for fractional PDE and their numerical approximation, Numerical Analysis and Its Applications, Lecture Notes in Comput. Sci. 8236, Springer, Heidelberg (2013), 38–49. 10.1007/978-3-642-41515-9_4Search in Google Scholar

[15] R. Lazarov and P. Vabishchevich, A numerical study of the homogeneous elliptic equation with fractional boundary conditions, Fract. Calc. Appl. Anal. 20 (2017), no. 2, 337–351. 10.1515/fca-2017-0018Search in Google Scholar

[16] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972. 10.1007/978-3-642-65161-8Search in Google Scholar

[17] C. Lubich, Fractional linear multistep methods for Abel–Volterra integral equations of the second kind, Math. Comp. 45 (1985), no. 172, 463–469. 10.1090/S0025-5718-1985-0804935-7Search in Google Scholar

[18] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704–719. 10.1137/0517050Search in Google Scholar

[19] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (Udine 1996), CISM Courses and Lectures 378, Springer, Vienna (1997), 291–348. 10.1007/978-3-7091-2664-6_7Search in Google Scholar

[20] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College, London, 2010. 10.1142/p614Search in Google Scholar

[21] V. Makarov, On a priori estimates of difference schemes giving an account of the boundary effect, C. R. Acad. Bulgare Sci. 42 (1989), no. 5, 41–44. Search in Google Scholar

[22] V. Makarov and N. Mayko, The boundary effect in the accuracy estimate for the grid solution of the fractional differential equation, Comput. Methods Appl. Math. 19 (2019), no. 2, 379–394. 10.1515/cmam-2018-0002Search in Google Scholar

[23] V. L. Makarov and L. I. Demkiv, Accuracy estimates of difference schemes for quasi-linear parabolic equations taking into account the initial-boundary effect, Comput. Methods Appl. Math. 3 (2003), no. 4, 579–595. 10.2478/cmam-2003-0036Search in Google Scholar

[24] V. L. Makarov and L. I. Demkiv, Accuracy estimates of difference schemes for quasi-linear elliptic equations with variable coefficient taking into account boundary effect, Numerical Analysis and Its Applications—NAA 2004, Lecture Notes in Comput. Sci. 3401, Springer, Berlin (2005), 80–90. 10.1007/978-3-540-31852-1_8Search in Google Scholar

[25] V. L. Makarov and L. I. Demkiv, Taking into account the third kind conditions in weight estimates for difference schemes, Large-Scale Scientific Computing, Lecture Notes in Comput. Sci. 3743, Springer, Berlin (2006), 687–694. 10.1007/11666806_79Search in Google Scholar

[26] A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer Briefs Appl. Sci. Technol., Springer, Cham, 2015. 10.1007/978-3-319-14756-7Search in Google Scholar

[27] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College, London, 2012. 10.1142/p871Search in Google Scholar

[28] N. V. Mayko, The boundary effect in the error estimate of the finite-difference scheme for the two-dimensional heat equation, J. Comput. Appl. Math. 3(113) (2013), 91–106. Search in Google Scholar

[29] N. V. Mayko, Error estimates of the finite-difference scheme for a one-dimensional parabolic equation with allowance for the effect of initial and boundary conditions, Cybernet. Systems Anal. 50 (2014), no. 5, 788–796. 10.1007/s10559-014-9669-6Search in Google Scholar

[30] N. V. Mayko, Improved accuracy estimates of the difference scheme for the two-dimensional parabolic equation with regard for the effect of initial and boundary conditions, Cybernet. Systems Anal. 53 (2017), no. 1, 83–91. 10.1007/s10559-017-9909-7Search in Google Scholar

[31] N. V. Mayko and V. L. Ryabichev, Boundary effect in the error estimate of the finite-difference scheme for two-dimensional Poisson’s equation, Cybernet. Systems Anal. 53 (2017), no. 1, 83–91. 10.1007/s10559-016-9877-3Search in Google Scholar

[32] R. V. Mendes and L. Vazquez, The dynamical nature of a backlash system with and without fluid friction, Nonlinear Dynam. 47 (2007), 363–366. 10.1007/s11071-006-9035-ySearch in Google Scholar

[33] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Search in Google Scholar

[34] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), no. 4, 367–386. Search in Google Scholar

[35] Y. Qin, Integral and Discrete Inequalities and Their Applications. Vol. I: Linear inequalities, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-33304-5Search in Google Scholar

[36] S. Saha Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics, CRC Press, Boca Raton, 2016. Search in Google Scholar

[37] A. A. Samarskii, The Theory of Difference Schemes, Nauka, Moscow, 1977. Search in Google Scholar

[38] A. A. Samarskii, R. D. Lazarov and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions (in Russian), Vysshaya Shkola, Moscow, 1987. Search in Google Scholar

[39] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993. Search in Google Scholar

[40] V. E. Tarasov, Fractional Dynamics, Nonlinear Phys. Sci., Springer, Heidelberg, 2010. 10.1007/978-3-642-14003-7Search in Google Scholar

[41] V. V. Vasil’ev and L. A. Simak, Fractional Calculus and Approximation Methods in Modelling of Dynamical Systems, National Academy of Sciences of Ukraine, Kyiv, 2008. Search in Google Scholar

[42] L. A. Ying and C. H. Wang, The discontinuous initial value problem of a reacting gas flow, Trans. Amer. Math. Soc. 266 (1981), no. 2, 361–387. 10.1090/S0002-9947-1981-0617539-9Search in Google Scholar

[43] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University, Oxford, 2008. Search in Google Scholar