# zbMATH — the first resource for mathematics

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation. (English) Zbl 1262.34008
Summary: We investigate the fractional differential equation $u^{\prime\prime}+A^c D^\alpha u=f(t, u,^c D^\mu u, u^\prime)$ subject to the boundary conditions $u^\prime(0)=0, u(T)+au^\prime(T)=0.$ Here, $$\alpha\in (1,2),\mu\in (0,1)$$, $$f$$ is a Carathéodory function and $$^c D$$ is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.

##### MSC:
 34A08 Fractional ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] Al-Mdallal Q.M., Syam M.I., Anwar M.N., A collocation-shooting method for solving fractional boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(12), 3814-3822 http://dx.doi.org/10.1016/j.cnsns.2010.01.020 · Zbl 1222.65078 [2] Çenesiz Y., Keskin Y., Kurnaz A., The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, J. Franklin Inst., 2010, 347(2), 452-466 http://dx.doi.org/10.1016/j.jfranklin.2009.10.007 · Zbl 1188.65107 [3] Coputo M., Linear models of dissipation whose Q is almost frequency independent. II, Fract. Calc. Appl. Anal., 2008, 11(1), 4-14 [4] Daftardar-Gejji V., Jafari H., Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 2005, 301(2), 508-518 http://dx.doi.org/10.1016/j.jmaa.2004.07.039 · Zbl 1061.34003 [5] Deimling K., Nonlinear Functional Analysis, Springer, Berlin, 1985 http://dx.doi.org/10.1007/978-3-662-00547-7 · Zbl 0559.47040 [6] Diethelm K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., 2004, Springer, Berlin, 2010 [7] Diethelm K., Ford N.J., Numerical solution of the Bagley-Torvik equation, BIT, 2002, 42(3), 490-507 · Zbl 1035.65067 [8] Edwards J.T., Ford N.J., Simpson A.C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. Comput. Appl. Math., 2002, 148(2), 401-418 http://dx.doi.org/10.1016/S0377-0427(02)00558-7 · Zbl 1019.65048 [9] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer, Berlin-New York, 1989 [10] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006 http://dx.doi.org/10.1016/S0304-0208(06)80001-0 [11] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999 · Zbl 0924.34008 [12] Raja M.A.Z., Khan J.A., Qureshi I.M., Solution of fractional order system of Bagley-Torvik equation using evolutionary computational intelligence, Math. Probl. Eng., 2011, #675075 · Zbl 1382.34010 [13] 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 http://dx.doi.org/10.1016/j.amc.2004.09.006 · Zbl 1109.65072 [14] Torvik P.J., Bagley R.L., On the appearance of the fractional derivative in the behavior of real materials, Trans. ASME J. Appl. Mech., 1984, 51(2), 294-298 http://dx.doi.org/10.1115/1.3167615 · Zbl 1203.74022 [15] Wang Z.H., Wang X., General solution of the Bagley-Torvik equation with fractional-order derivative, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(5), 1279-1285 http://dx.doi.org/10.1016/j.cnsns.2009.05.069 · Zbl 1221.34020
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.