Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering.

中文翻译：

---

分数阶移位勒让德tau方法求解线性和非线性变量阶分数阶偏微分方程

在这里，我们通过分数位移勒让德Tau方法揭示了分数阶线性和非线性Klein-Gorden偏微分方程。以此为目标，导出分数阶移位勒让德函数（FSLF）的运算矩阵，并将其与Tau方法组合，以将分数阶微分方程转换为可解代数方程组。测试了运算矩阵的有效性和效率。我们的发现产生了肯定的结果，表明所提出的方法适用于科学和工程中出现的非线性方程。