Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

Abstract

The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is 1 < α < 2, the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is 0 < α < 1, the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.

Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models

“…The Bagley–Torvik equation has the same structure as equation (5) and is widely studied in the literature (Arqub and Maayah, 2018; Daftardar-Gejji and Jafari, 2005; Diethelm and Ford, 2002; Huang et al, 2016; Wang and Wang, 2010). An iterative method for solving such a problem can be found in Arqub and Maayah (2018), and the following example is chosen from this paper.…”

Existence and Hyers–Ulam stability for three-point boundary value problems with Riemann–Liouville fractional derivatives and integrals

“…For example, Cermak et al [12] investigated the twoterm fractional differential equation The analytic solution and the numerical solution for the B-T equation were studied in [13] and [14], respectively. Various methods were introduced to investigate the approximate solutions such as the finite difference method [15], the variational iteration method [13,16], the homotopy perturbation method [17] and the generalized differential transform method [18]. Boundary value problems for the B-T equations were studied in [19,20] and [18,[21][22][23] for various boundary value conditions.…”

“…Various methods were introduced to investigate the approximate solutions such as the finite difference method [15], the variational iteration method [13,16], the homotopy perturbation method [17] and the generalized differential transform method [18]. Boundary value problems for the B-T equations were studied in [19,20] and [18,[21][22][23] for various boundary value conditions. In [18], the authors considered the approximate solution of B-T equations with variable coefficients and three-point boundary value,…”

“…Boundary value problems for the B-T equations were studied in [19,20] and [18,[21][22][23] for various boundary value conditions. In [18], the authors considered the approximate solution of B-T equations with variable coefficients and three-point boundary value,…”