In this article, an efficient variable‐order Chebyshev collocation method which is based on shifted fifth‐kind Chebyshev polynomials is applied to solve a nonlinear variable‐order fractional reaction–diffusion equation with Mittag–Leffler kernel. The operational matrix of shifted fifth‐kind Chebyshev polynomials is derived for variable‐order ABC derivatives. The Chebyshev operational matrix together with the collocation method are applied to concerned nonlinear physical model with Mittag–Leffler kernel which is converted into a system of nonlinear algebraic equations, this system can be solved by using Newton method. The main focus of this paper is finding the convergence analysis of the approximation and high convergence order for small grid approximation. Few test examples with a comparison of maximum absolute error between the obtained numerical solution and existing known solution are being reported to show the accuracy and stability of the scheme. The physical presentation of the absolute errors for considered nonlinear variable‐order reaction–diffusion equations involving the Mittag–Leffler kernel with their exact solutions shows that the method is good for finding the solution of these kind of problems.

中文翻译：

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一类带有Mittag-Leffler核的非线性变阶分数阶反应扩散方程的解

在本文中，基于移位的第五种Chebyshev多项式的一种有效的变阶Chebyshev配置方法被用于求解带有Mittag-Leffler核的非线性变阶分数阶反应-扩散方程。对于可变阶ABC导数，得出了移位的第五类Chebyshev多项式的运算矩阵。将Chebyshev运算矩阵与搭配方法一起应用到具有Mittag-Leffler核的相关非线性物理模型中，该模型被转换为非线性代数方程组，可以使用牛顿法求解该系统。本文的主要重点是找到小网格逼近的逼近和高收敛阶的收敛分析。很少有测试示例可以比较所获得的数值解和现有已知解之间的最大绝对误差，以显示该方案的准确性和稳定性。对涉及Mittag-Leffler核的非线性变阶反应扩散方程的绝对误差的物理表示及其精确解法表明，该方法可很好地找到此类问题的解。