On a Generalized Bagley–Torvik Equation with a Fractional Integral Boundary Condition

The Bagley–Torvik equation is generalized by using variable coefficients and the fractional order \(0<\alpha <2.\) A fractional integral boundary condition is proposed to investigate the nonlocal behavior of the generalized Bagely–Torvik equation. Based on the concept of Riemann–Liouville fractional derivative, the Fredholm integral equations of the second kind are derived for \(0<\alpha <1\) and \(1\le \alpha <2\), respectively. Moreover, the generalized piecewise Taylor-series expansion method is proposed to find the approximate solution, and its convergence and error estimate are analyzed. Numerical results are reported to illustrate the effects of the boundary-value conditions on the approximate solutions. The obtained results reveal that the boundary value conditions play an important role to appropriately model a practical process.