This article presents a novel computational algorithm for solving the fractional pantograph differential equation (FPDE). The algorithm is based on introducing a new family of orthogonal polynomials, generalizing the second-kind Chebyshev polynomials family. Specifically, we use the shifted generalized Chebyshev polynomials of the second kind (SGCPs) as basis functions, approximating the solutions as linear combinations of these new polynomials. First, we establish new theoretical results related to these polynomials, which form the foundation of our proposed method. Then, we apply the spectral Galerkin method to convert the FPDE with its initial conditions into an algebraic system that can be numerically solved. Additionally, we analyze the convergence of the proposed expansion. Finally, numerical examples are provided to validate the theoretical results.

中文翻译：

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使用某些正交广义切比雪夫多项式的分数延迟受电弓方程的新谱算法


本文提出了一种用于求解分数受电弓微分方程 （FPDE） 的新型计算算法。该算法基于引入一个新的正交多项式族，推广了第二类切比雪夫多项式族。具体来说，我们使用第二类移位广义切比雪夫多项式 （SGCPs） 作为基函数，将解近似为这些新多项式的线性组合。首先，我们建立了与这些多项式相关的新理论结果，这些结果构成了我们提出的方法的基础。然后，我们应用谱 Galerkin 方法将 FPDE 及其初始条件转换为可以数值求解的代数系统。此外，我们还分析了拟议扩展的收敛性。最后，给出了数值算例来验证理论结果。