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Abstract
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A fractional differential equation called the Bagley-Torvik equation is frequently used to simulate fluid-solid interactions, damped mechanical structures, and viscoelastic systems in situations where memory effects are not well captured by conventional integer-order models. There are rarely analytical solutions for these kinds of equations, and conventional numerical methods frequently have poor accuracy, instability, or efficiency, particularly for lengthy simulations or rigid system parameters. This emphasizes the need for extremely precise and computationally effective techniques that can manage fractional derivatives successfully. Extensive convergence, high-order approximation, and lower computational cost are just a few benefits of the suggested spectral approach employing Vieta–Lucas polynomial basis, which makes it possible to solve Bagley–Torvik equations precisely and steadily with fewer basis functions. This research is important because it offers a reliable and effective computational framework that improves the theoretical knowledge and real-world simulation of fractional viscoelastic and dynamical systems in physics, applied mathematics, and engineering.
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