Document Type : Research Article
Authors
Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq
Abstract
The Bagley--Torvik equation, which governs the motion of a rigid plate immersed in a Newtonian fluid with viscoelastic damping, represents one of the canonical benchmark problems in fractional calculus. This paper presents a systematic comparative numerical analysis of four established methods for solving this equation: the Fractional-Order Hybrid Jacobi Functions method (FOHJF), the Polynomial Least Squares Method (PLSM), the Vieta-Lucas Spectral Method (VLSM), and the Cubic Spline Collocation Method (CSCM). Three benchmark test problems with qualitatively distinct forcing functions---polynomial, oscillatory (cosine), and exponential---and varying initial conditions are used to evaluate absolute approximation errors at resolution levels N = 8, 16, 32, and 64. Detailed error tables and graphical convergence analyses are provided. The results consistently demonstrate that VLSM achieves the highest accuracy, with maximum absolute errors below 2.3×10⁻⁸ at N = 32, followed by FOHJF. PLSM and CSCM offer simpler implementation at the cost of reduced accuracy. Practical recommendations are provided for selecting a method based on the required precision level and the type of forcing function. The study identifies key limitations and directions for future work, including extension to nonlinear formulations, variable-order derivatives, and adaptive hybrid approaches.
Highlights
- Four numerical methods (VLSM, FOHJF, PLSM, CSCM) are systematically benchmarked for the Bagley–Torvik fractional equation.
- Three benchmark test problems with polynomial, oscillatory, and exponential exact solutions are used for evaluation.
- VLSM achieves the highest accuracy with maximum absolute errors below 2.3×10⁻⁸ at N = 32.
- Convergence studies across resolutions N = 8, 16, 32, 64 confirm exponential convergence of spectral methods.
- Practical guidelines for selecting methods are provided based on the required accuracy and the forcing function.
Keywords
- Bagley--Torvik equation
- Caputo fractional derivative
- Spectral methods
- Collocation methods
- Vieta-Lucas polynomials
- Fractional-order Jacobi functions
- Error analysis
Main Subjects
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