A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley-Torvik Differential Equation

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley-Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss-Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme's accuracy and compare it to existing findings.