Boundary Characteristic Orthogonal Polynomials-Based Galerkin and Least Square Methods for Solving Bagley–Torvik Equations

Abstract

In this paper, efficient numerical methods for solving Bagley–Torvik (B-T) equations with variable coefficients and three-point boundary value conditions are considered. This model is considered as a viscoelastic behavior of geological strata, metal, and glasses using fractional differential equations. Many viscoelastic materials are proposed in which derivatives of fractional-order replace the usual time derivatives of integer order. An application of such a model is the prediction of the transient response of frequency-dependent materials. As such the titled problem is challenging to solve using the efficient method(s). The fractional derivative is described in the Caputo sense. First, a linearly independent set such as $\left\{ {1,x,x^{2} ,x^{3} , \ldots } \right\}$ is converted to Boundary Characteristic Orthogonal Polynomials (BCOPS) by Gram–Schmidt Orthogonalization process then these are used in the Galerkin and Least Square methods to reduce B-T Equations to the linear or nonlinear system of algebraic equations. Example problems are addressed to show the powerfulness and efficacy of the method.