An investigation of fractional Bagley-Torvik equation. (English) Zbl 1425.34010

Summary: In this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.


34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text: DOI


[1] Diethelm K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010
[2] Kilbas A. A., Srivastava H. M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, in : North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006 · Zbl 1092.45003
[3] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999 · Zbl 0918.34010
[4] Sabatier J., Agrawal O.P., Machado J. A. T., Advances in Fractional Calculus : Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007 · Zbl 1116.00014
[5] Zaslavsky G. M., Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005 · Zbl 1083.37002
[6] Magin R. L., Fractional Calculus in Bioengineering, Begell House Publisher, Inc., Connecticut, 2006
[7] Fazli H., Bahrami F., On the steady solutions of fractional reaction-diffusion equations, Filomat, 2017, 31(6), 1655-1664 · Zbl 1499.35635
[8] Bahrami F., Fazli H., Jodayree Akbarfam A., A new approach on fractional variational problems and Euler-Lagrange equations, Commun. Nonlinear Sci. Numer. Simul. 2015, 23(13), 39-50 · Zbl 1351.49020
[9] Fazli H., Nieto J. J., Nonlinear sequential fractional differential equations in partially ordered spaces, Filomat, 32 (2018), 4577-4586. · Zbl 1513.34021
[10] Fazli H., Nieto J. J., Fractional Langevin equation with anti-periodic boundary conditions, Chaos, Solitons and Fractals, 114 (2018) 332-337. · Zbl 1415.34016
[11] Fazli H., Nieto J .J., F. Bahrami, On the existence and uniqueness results for nonlinear sequential fractional differential equations, Appl. Comput. Math., 17 (2018) 36-47. · Zbl 1453.34006
[12] Kadem A., Baleanu D., Solution of a fractional transport equation by using the generalized quadratic form, Commun. Nonlinear Sci. Numer. Simul., 2011, 16, 3011-3014 · Zbl 1220.45006
[13] Singh J., Kumar D., Nieto J. J., Analysis of an El Nino-Southern Oscillation model with a new fractional derivative, Chaos, Solitons and Fractals, 2017, 99, 109-115 · Zbl 1373.86007
[14] Sadallah M., Muslih S. I., Baleanu D., Rabei E., Fractional time action and perturbed gravity, Fractals, 2011, 19, 243-247 · Zbl 1222.28019
[15] Din X. L., Nieto J. J., Controllability of nonlinear fractional delay dynamical systems with prescribed controls, Nonlinear Analysis : Modelling and Control, 2018, 23, 1-18 · Zbl 1416.93028
[16] Lazarevic M. P., Spasic A. M., Finite-time stability analysis of fractional order time delay systems : Gronwall’s approach, Math. Comput. Modelling., 2009, 49, 475-481 · Zbl 1165.34408
[17] Bonilla B., Rivero M., Rodriguez-Germa L., Trujillo J. J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 2007, 187, 79-88 · Zbl 1120.34323
[18] Torvik P. J., Bagley R. L., On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech., 1984, 51, 294-298 · Zbl 1203.74022
[19] Bagley R. L., Torvik P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 1983, 27, 201-210 · Zbl 0515.76012
[20] Torvik P. J., Bagley R. L., Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J., 1985, 23, 918-925 · Zbl 0562.73071
[21] Diethelm K., Ford N. J., Numerical solution of the Bagley-Torvik equation, BIT Numerical Mathematics, 2002, 42, 490-507 · Zbl 1035.65067
[22] Çenesiz Y., Keskin Y., Kurnaz A., The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, J. Frankl. Inst., 2010, 347, 452-466 · Zbl 1188.65107
[23] Karaaslan M. F., Celiker F., Kurulay M., Approximate solution of the Bagley-Torvik equation by hybridizable discontinuous Galerkin methods, Appl. Math. Comput., 2013, 219 (11), 6328-6343
[24] Zahra W. K., Van Daele M., Discrete spline methods for solving two point fractional Bagley-Torvik equation, Appl. Math. Comput., 2017, 296, 42-56 · Zbl 1411.65098
[25] Mekkaoui T., Hammouch Z., Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method, Ann. Univ. Craiova Math. Comput. Sci. Ser., 2012, 39 (2), 251-256 · Zbl 1274.34010
[26] Mohammadi F., Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative, Int. J. Adv. in Appl. Math. and Mech., 2014, 2 (1), 83-91 · Zbl 1359.65136
[27] Ray S. S., Bera R. K., Analytical solution of the Bagley-Torvik equation by Adomian decomposition method, Appl. Math. Comput., 2005, 168 (1), 398-410 · Zbl 1109.65072
[28] Ray S. S., On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley-Torvik equation, Appl. Math. Comput., 2012, 218, 5239-5248 · Zbl 1359.65314
[29] Staněk S., Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation, Cent. Eur. J. Math., 2013, 11(3), 574-593 · Zbl 1262.34008
[30] Wang Z. H., Wang X., General solution of the Bagley-Torvik equation with fractional-order derivative, Commun Nonlinear Sci Numer Simulat., 2010, 15, 1279-1285 · Zbl 1221.34020
[31] Ran A. C. M., Reurings M. C. B., Fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 2003, 132, 1435-1443 · Zbl 1060.47056
[32] Nieto J. J., Rodríguez-López R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 2005, 22, 223-239 · Zbl 1095.47013
[33] Nieto J. J., Rodríguez-López R., Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 2007, 135, 2505-2517 · Zbl 1126.47045
[34] Bhaskar T. G., Lakshmikantham V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 2006, 65, 1379-1393 · Zbl 1106.47047
[35] Luong N. V., Thuan N. X., Coupled fixed point theorems for mixed monotone mappings and an application to integral equations, Computers and Mathematics with Applications, 2011, 62, 4238-4248 · Zbl 1236.47056
[36] Hardy G. H., Littlewood J. E., Some properties of fractional integrals. I, Math. Z., 1928, 27, 565-606 · JFM 54.0275.05
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.