In this paper, we study and present a spectral numerical technique for solving a general class of multi-order fractional pantograph equations with varying coefficients and systems of pantograph equations. In this study, the spectral Galerkin approach in combination with the properties of shifted Legendre polynomials is used to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with fractional pantograph equations via spectral Galerkin approach. The errors and convergence of the adopted approach are rigorously analyzed. The efficiency and accuracy of the technique are tested by considering five different examples, to ensure that the suggested approach is more accurate than the existing other techniques. The obtained results in this paper are comparing favorably with those published by other researchers and with the existing exact solutions, whenever possible.

在本文中，我们研究并提出了一种频谱数值技术，用于求解一类通用的具有变化系数的多阶分数阶受电弓方程和受电弓方程组。在这项研究中，频谱Galerkin方法与移位的Legendre多项式的性质相结合，可以将此类方程简化为代数方程组，可以使用任何合适的求解器进行求解。据作者所知，这是通过谱Galerkin方法处理分数缩放弓方程的首次尝试。严格分析了所采用方法的误差和收敛性。通过考虑五个不同的示例来测试该技术的效率和准确性，以确保所建议的方法比现有的其他技术更准确。