论文标题
双变量分形插值在三角形结构域上的数值整合和近似值
Bivariate fractal interpolation functions on triangular domain for numerical integration and approximation
论文作者
论文摘要
本文的主要目的是使用顶点着色的概念在三角形插值域上构建双变量分形插值函数，并为构造的插值函数提出双重整合公式。与传统的构造不同，三角区域分区中的每个顶点已经分配了一种颜色，使得分区的色数为3。提出了三角形分配的新方法，并与其图形的变色数有关。构造后，提供了确定垂直缩放系数的公式。通过新定义的垂直缩放因子，可以清楚地观察到双重积分的值与使用分形理论计算的积分值一致。此外，建立了将分形插值函数与通过三角形顶点的平面方程连接的关系。通过足够的引理和定理证明了所提出的方法与实际积分值的收敛性。还提供了足够的例子来说明构建方法并验证双重整合的公式。
The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided. With the newly defined vertical scaling factor, it is clearly observed that the value of the double integral coincides with the integral value calculated using fractal theory. Further, a relation connecting the fractal interpolation function with the equation of the plane passing through the vertices of the triangle is established. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Sufficient examples are also provided to illustrate the method of construction and to verify the formula of double integration.
