The calculus of the mixed Weyl–Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl–Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl–Marchaud fractional derivative having fractional order γ = (p,q) of a continuous function which satisfies μ-Hölder condition is no more than 3 − μ + (p + q) when 0 < p, q < μ < 1, p + q < μ, which reveals an important phenomenon about linearly increasing effect of dimension of the mixed Weyl–Marchaud fractional derivative. Furthermore, we deduce box dimension of the graph of the mixed Weyl–Marchaud fractional derivative of a continuous function which is defined on a rectangular region in ℝ2 and also, we analyze that the mixed Weyl–Marchaud fractional derivative of a function preserves some basic properties such as continuity, bounded variation and boundedness. The results are new and compliment the existing ones.

中文翻译：

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混合WEYL-MARCHAUD分数导数和盒维数的分析

本文研究了混合Weyl-Marchaud分数导数的微积分。我们证明二元分形插值函数（FIF）的混合Weyl-Marchaud分数导数仍然是二元FIF。证明了具有分数阶的混合Weyl-Marchaud分数导数的上盒维数γ = (p,q)一个满足的连续函数μ-Hölder 条件不超过3 - μ + (p + q)什么时候0 < p,q < μ < 1,p + q < μ，这揭示了混合Weyl-Marchaud分数导数的维数线性增加效应的重要现象。此外，我们推导出定义在矩形区域上的连续函数的混合 Weyl-Marchaud 分数导数图的框维数ℝ2此外，我们分析了函数的混合 Weyl-Marchaud 分数导数保留了一些基本性质，例如连续性、有界变化和有界性。结果是新的，并补充了现有的结果。