Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

We investigate the fractional differential equation u″ + AcDαu = f(t, u, cDμu, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and cD 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.

[1]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[2]  Junaid Ali Khan,et al.  Solution of Fractional Order System of Bagley-Torvik Equation Using Evolutionary Computational Intelligence , 2011 .

[3]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[4]  Hossein Jafari,et al.  Adomian decomposition: a tool for solving a system of fractional differential equations , 2005 .

[5]  Santanu Saha Ray,et al.  Analytical solution of the Bagley Torvik equation by Adomian decomposition method , 2005, Appl. Math. Comput..

[6]  M. Anwar,et al.  A collocation-shooting method for solving fractional boundary value problems , 2010 .

[7]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[8]  F. Browder.  Nonlinear functional analysis , 1970 .

[9]  N. Ford,et al.  The numerical solution of linear multi-term fractional differential equations: systems of equations , 2002 .

[10]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[11]  Z. Wang,et al.  General solution of the Bagley–Torvik equation with fractional-order derivative , 2010 .

[12]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[13]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[14]  Aydin Kurnaz,et al.  The solution of the Bagley-Torvik equation with the generalized Taylor collocation method , 2010, J. Frankl. Inst..

[15]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .