In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time direction, the fractal–fractional derivative in the Atangana–Riemann–Liouville concept is utilized whereas the fractional derivatives in the Caputo and Atangana–Baleanu–Caputo senses are mutually used in the space variable to define this new class of problems. The presented method utilizes the Bernstein polynomials (BPs) and their operational matrices of fractional and fractal–fractional derivatives (which are generated in this study). To this end, the unknown solution is expanded by the BP and is replaced in the equation. Then, the generated operational matrices and the collocation method are employed to generate a system of algebraic equations. Eventually, by solving this system a numerical solution is obtained for the problem. The validity of the designed method is investigated through three numerical examples.

中文翻译：

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非线性平流-扩散-反应方程的时空分形-分形模型通过伯恩斯坦多项式的数值处理

本文介绍了非线性时空分形-分数平流-扩散-反应方程，并提出了一种高精度的数值求解方法。在时间方向上，利用了 Atangana-Riemann-Liouville 概念中的分形-分数导数，而 Caputo 和 Atangana-Baleanu-Caputo 意义上的分数导数在空间变量中相互使用来定义这类新问题。所提出的方法利用伯恩斯坦多项式 (BP) 及其分数和分形分数导数的运算矩阵（在本研究中生成）。为此，未知解由BP扩展并在方程中替换。然后，使用生成的运算矩阵和搭配方法来生成代数方程组。最终，通过求解该系统，得到了该问题的数值解。通过三个数值例子研究了所设计方法的有效性。