Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 Articles in Press 2026 05 28 Comparative Accuracy Analysis of Spectral and Collocation Methods for the Fractional Bagley–Torvik Equation: A Systematic Numerical Study 12983 10.30473/coam.2026.77807.1408 EN Muayyad Mahmood Khalil Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq 0009-0005-3482-3034 Najim Abdullah Ibrahim Department of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq Journal Article 2026 02 22 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 <em>N</em> = 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 <em>N</em> = 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. https://mathco.journals.pnu.ac.ir/article_12983_a5dd299d95ff0fbc9b66534d41ab565f.pdf