Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains

The paper exhibits a practical and effective scheme to approximate the solutions of a class of fractional differential equations known as the Bagley-Torvik equations. The underlying fractional derivative is based on the Caputo definition. Both the boundary and initial conditions are considered while the domain of approximation is taken sufficiently large. The principle idea behind our approximation algorithm is to use a novel class of (orthogonal) polynomials called the Bessel polynomials, which their coefficients are positive. The convergence and error analysis of the Bessel solution are investigated in the L-2 and infinity norms. Representing the unknown solution and its derivatives in terms of basis functions together with collocation points, the Bagley-Torvik equations are reduced into an algebraic form. In particular, we present a simple but fast algorithm with linear complexity for computing the q-th order fractional derivative of the basis functions in the vectorized form. Several practical test problems are given to illustrate the utility and efficiency of the proposed approximation algorithm. Validation of the method is obtained by comparison with existing available numerical solutions. Based on experiments, the present approach produces the results of high accuracy to obtain the approximate solutions of Bagley-Torvik equations especially for long-time computations.

Automated summary

This paper presents a practical and effective scheme to approximate the solutions of the Bagley-Torvik equations using Bessel polynomials as approximation basis functions. The equations are reduced into an algebraic form and a fast algorithm with linear complexity is introduced to compute the fractional derivatives. The proposed approximation algorithm shows high accuracy and efficiency for long-time computations.