A numerical scheme utilizing shifted Bernstein polynomials is developed to address the variable fractional-order governing equation in viscoelastic fluid-conveying pipes. The pipe’s mechanical response is characterized through a variable fractional-order Kelvin–Voigt (FKV) model, which effectively captures the time-dependent and memory properties of viscoelastic materials. By coupling the FKV constitutive model with the motion equation, the governing equation for a viscoelastic pipe is obtained. The deformation field is approximated using shifted Bernstein trial functions, leading to the construction of a derivative matrix with a variable fractional order. The obtained governing relation is expressed in matrix form, and after discretization, an algebraic system is formulated that is solvable in the time domain to evaluate the pipe displacement. Moreover, a convergence investigation is carried out to examine the reliability and effectiveness of the proposed framework. Computational results demonstrate that the introduced method delivers outstanding precision and performance, while the viscoelastic pipe’s response under different scenarios—including applied loads and fluid flow rates—is comprehensively investigated.

This study presents a shifted Bernstein polynomial-based method for numerically solving the variable fractional order control equation governing a viscoelastic bar. Initially, employing a variable order fractional constitutive relation alongside the equation of motion, the control equation for the viscoelastic bar is derived. Shifted Bernstein polynomials serve as basis functions for approximating the bar’s displacement function, and the variable fractional derivative operator matrix is developed. Subsequently, the displacement control equation of the viscoelastic bar is transformed into the form of a matrix product. Substituting differential operators into the control equations, the control equations are discretized into algebraic equations by the method of matching points, which in turn allows the numerical solution of the displacement of the variable fractional viscoelastic bar control equation to be solved directly in the time domain. In addition, a convergence analysis is performed. Finally, algorithm precision and efficacy are confirmed via computation.