A high-precision spectral method for solving Bagley–Torvik equations

The analytical solution of Bagley–Torvik equations involves a convolution integral that includes the Green’s function and the source term. However, when the source term is a general function rather than a polynomial, the computation becomes challenging. Furthermore, if the problem is governed by nonhomogeneous initial conditions, the resulting convolution integral contains more complex expressions, such as multivariate generalizations of the Mittag–Leffler function, which are difficult to evaluate. From a numerical perspective, traditional methods such as those based on operational matrices (involving both derivative and integral operational matrices), as well as spectral Tau and collocation techniques, are commonly used to solve Bagley–Torvik equations. However, these approaches tend to be computationally intensive, less user-friendly, and may become unstable when a large number of orthogonal polynomial terms are employed. To address these challenges, this study proposes a novel and efficient computational technique from the class of spectral methods to solve Bagley–Torvik equations. The proposed method is capable of addressing both homogeneous and nonhomogeneous initial conditions and accommodates a broad class of source terms. It features a newly developed fractional-order integral operational matrix based on Chelyshkov polynomials, which efficiently transforms the Bagley–Torvik equations into a system of algebraic equations. This streamlined formulation eliminates the need for derivative operational matrices as well as spectral Tau and collocation techniques. The method’s effectiveness is validated through comparisons with exact solutions, Podlubny’s numerical results, and several existing analytical and numerical approaches. In the absence of exact solutions, additional validation is performed using benchmark results from Podlubny’s MATLAB’s repository. It is worth mentioning that the proposed method is equally suitable for solving nonlinear fractional differential equations. Overall, the proposed approach offers enhanced accuracy, reduced computational complexity, and faster convergence, establishing it as a highly efficient tool for solving Bagley–Torvik equations.