The research object of this paper is the mixed $(\kappa ,s)$-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed $(\kappa ,s)$-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under $\sigma =({\sigma }_1,{\sigma }_2)$ order of the mixed integral is $3-min\{\frac{{\sigma }_1}{\kappa },\frac{{\sigma }_2}{\kappa }\}$ where $\kappa >0$.

本文的研究对象是混合$（\kappa ,s）$-矩形区域上二元函数的黎曼-刘维尔分数积分，这是单变量函数分数积分的自然推广。本文首先指出混合积分仍然保持了经典性质的有效性，如有界性、连续性和有界变分等。此外，我们研究了混合积分下双变量函数的分形维数，包括 Hausdorff 维数和 Box 维数。主要结果表明混合图的分形维数$（\kappa ,s）$-具有有界变分的连续函数的黎曼-刘维尔积分仍然是二。二维连续函数混合积分的 Box 维数也已计算。此外，我们证明了二元连续函数的 Box 维数的上界$\sigma =（{\sigma }_1,{\sigma }_2）$混合积分的阶数为$3-分钟\{\frac{{\sigma }_1}{\kappa },\frac{{\sigma }_2}{\kappa }\}$在哪里 $\kappa >0$。