Abstract In this paper, we generalize a coupled system of nonlinear reaction-advection-diffusion equations to a variable-order fractional one by using the Caputo-Fabrizio fractional derivative, which is a non-singular fractional derivative operator. In order to establish an appropriate method for this system, we introduce a new formulation of the discrete Legendre polynomials namely the orthonormal shifted discrete Legendre polynomials. The operational matrices of classical and fractional derivatives of these basis functions are extracted. The devised method uses these polynomials and their operational matrices together with the collocation technique to transform the system under consideration into a system of algebraic equations which is uncomplicated for solving. Two numerical examples are analyzed to examine the accuracy of the method.

中文翻译：

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用于求解非线性变阶时间分数阶反应-平流-扩散方程耦合系统的正交移位离散勒让德多项式

摘要 本文利用非奇异分数阶导数算子Caputo-Fabrizio分数阶导数将非线性反应-平流-扩散方程的耦合系统推广到变阶分数阶方程。为了为该系统建立合适的方法，我们引入了离散勒让德多项式的新公式，即正交移位离散勒让德多项式。提取这些基函数的经典和分数阶导数的运算矩阵。所设计的方法使用这些多项式及其运算矩阵以及搭配技术将所考虑的系统转换为求解不复杂的代数方程组。分析了两个数值例子来检验该方法的准确性。