Giải pháp phân tích của phương trình Bagley–Torvik tổng quát

Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Academic Press, Amsterdam (2016) Caputo, M., Mainardi, F.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015) Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Academic Press, Amsterdam (2016) Hristov, J.: Linear viscoelastic responses: the prony decomposition naturally leads into the Caputo–Fabrizio fractional operator. Front. Phys. 6, 135 (2018) Yang, X.J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2018) Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) Hristov, J.: On the Atangana–Baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation. In: Gómez, J., Torres, L., Escobar, R. (eds.) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control. Springer, Cham (2019) Yang, X.J., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Spec. Top. 226(16–18), 3567–3575 (2017) Cattani, C.: Sinc-fractional operator on Shannon wavelet space. Front. Phys. 6, 118 (2018) Sun, L., Chen, L.: Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives. J. Sound Vib. 335(20), 19–33 (2015) Hristov, J.: Response functions in linear viscoelastic constitutive equations and related fractional operators. Math. Model. Nat. Phenom. 14(3), 305 (2019) Yang, X.J., Gao, F., Srivastava, H.M.: New rheological models within local fractional derivative. Rom. Rep. Phys. 69, 113 (2017) Yang, X.J.: New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 19(1), 45–52 (2018) Jiang, Y., Wang, X., Wang, Y.: On a stochastic heat equation with first order fractional noises and applications to finance. J. Math. Anal. Appl. 396(2), 656–669 (2012) Baleanu, D., Jajarmi, A., Bonyah, E., Hajipour, M.: New aspects of poor nutrition in the life cycle within the fractional calculus. Adv. Differ. Equ. 2018, 230 (2018) Alkahtani, B.S.T.: Chua’s circuit model with Atangana–Baleanu derivative with fractional order. Chaos Solitons Fractals 89, 547–551 (2016) Pang, D.H., Jiang, W., Liu, S., Du, J.: Stability analysis for a single degree of freedom fractional oscillator. Physica A 523, 498–506 (2019) Jajarmi, A., Hajipour, M., Mohammadzadeh, E., Baleanu, D.: A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. J. Franklin Inst. 335(9), 3938–3967 (2018) Zhang, H., Ye, M.L., Ye, R.Y., Cao, J.D.: Synchronization stability of Riemann–Liouville fractional delay-coupled complex neural networks. Physica A 508, 155–165 (2018) Yang, X.J., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20(2), 753–756 (2016) Yang, A.M., Han, Y., Li, J., Liu, W.X.: On steady heat flow problem involving Yang–Srivastava–Machado fractional derivative without singular kernel. Therm. Sci. 20(3), 717–721 (2016) Hristov, J.: Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo–Fabrizio time-fractional derivative. Therm. Sci. 20(2), 757–762 (2016) Yang, X.J., Machado, J.A.T., Baleanu, D.: Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom. Rep. Phys. 69, 115 (2017) Yang, X.J., Gao, F., Ju, Y., Zhou, H.W.: Fundamental solutions of the general fractional-order diffusion equations. Math. Methods Appl. Sci. 41, 9312–9320 (2018) Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Fractional, M.R.: Bloch equation with delay. Comput. Math. Appl. 61, 1355–1365 (2011) Agila, A., Baleanu, D., Eid, R., Irfanoglu, B.: Applications of the extended fractional Euler–Lagrange equations model to freely oscillating dynamical systems. Rom. J. Phys. 61(3), 350–359 (2016) Baghani, O.: On fractional Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simul. 42, 675–681 (2017) 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) Bagley, R.L.: On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fract. Calc. Appl. Anal. 10(2), 123–126 (2007) Slonimsky, G.L.: Laws of mechanical relaxation processes in polymers. J. Polym. Sci., C Polym. Symp. 16(3), 1667–1672 (1967) Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984) Pritz, T.: Five-parameter fractional derivative model for polymeric damping materials. J. Sound Vib. 265, 935–952 (2003) Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II) 1, 161–198 (1971) Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983) Pang, D.H., Jiang, W., Liu, S., Niazi, A.U.K.: Well-posedness and iterative formula for fractional oscillator equations with delays. Math. Methods Appl. Sci. 1–13 (2019). https://doi.org/10.1002/mma.5572 Dai, H.Z., Zheng, Z.B., Wang, W.: On generalized fractional vibration equation. Chaos Solitons Fractals 95, 48–51 (2017) Yuan, J., Zhang, Y.A., Liu, J.M., Shi, B., Gai, M.J., Yang, S.J.: Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators. J. Sound Vib. 397, 192–203 (2017) Mohamed, E.G., Mahmoud, A.E.: Numerical solution of the Bagley–Torvik equation by Legendre-collocation method. SeMA J. 74(4), 371–383 (2017) Stanĕk, S.: Two-point boundary value problems for the generalized Bagley–Torvik fractional differential equation. Cent. Eur. J. Math. 11(3), 574–593 (2013) Zhong, X.C., Liu, X.L., Liao, S.L.: On a generalized Bagley–Torvik equation with a fractional integral boundary condition. Int. J. Appl. Comput. Math. 3(1), 727–746 (2017) Labecca, W., Guimaraes, O., Piqueira, J.: Analytical solution of general Bagley–Torvik equation. Math. Probl. Eng. 2015, Article ID 591715 (2015) Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)